GHC/Core/FamInstEnv.hs

    {-# LANGUAGE CPP                 #-}
    {-# LANGUAGE DeriveFunctor       #-}
    {-# LANGUAGE GADTs               #-}
    {-# LANGUAGE ScopedTypeVariables #-}
    {-# LANGUAGE TupleSections       #-}

    {-# OPTIONS_GHC -Wno-incomplete-record-updates #-}
    -- (c) The University of Glasgow 2006
    --
    -- FamInstEnv: Type checked family instance declarations

    module GHC.Core.FamInstEnv (
            FamInst(..), FamFlavor(..), famInstAxiom, famInstTyCon, famInstRHS,
            famInstsRepTyCons, famInstRepTyCon_maybe, dataFamInstRepTyCon,
            pprFamInst, pprFamInsts,
            mkImportedFamInst,

            FamInstEnvs, FamInstEnv, emptyFamInstEnv, emptyFamInstEnvs,
            extendFamInstEnv, extendFamInstEnvList,
            famInstEnvElts, famInstEnvSize, familyInstances,

            -- * CoAxioms
            mkCoAxBranch, mkBranchedCoAxiom, mkUnbranchedCoAxiom, mkSingleCoAxiom,
            mkNewTypeCoAxiom,

            FamInstMatch(..),
            lookupFamInstEnv, lookupFamInstEnvConflicts, lookupFamInstEnvByTyCon,

            isDominatedBy, apartnessCheck,

            -- Injectivity
            InjectivityCheckResult(..),
            lookupFamInstEnvInjectivityConflicts, injectiveBranches,

            -- Normalisation
            topNormaliseType, topNormaliseType_maybe,
            normaliseType, normaliseTcApp,
            topReduceTyFamApp_maybe, reduceTyFamApp_maybe
        ) where

    #include "HsVersions.h"

    import GHC.Prelude

    import GHC.Core.Unify
    import GHC.Core.Type as Type
    import GHC.Core.TyCo.Rep
    import GHC.Core.TyCon
    import GHC.Core.Coercion
    import GHC.Core.Coercion.Axiom
    import GHC.Types.Var.Set
    import GHC.Types.Var.Env
    import GHC.Types.Name
    import GHC.Types.Unique.DFM
    import GHC.Data.Maybe
    import GHC.Types.Var
    import GHC.Types.SrcLoc
    import Control.Monad
    import Data.List( mapAccumL )
    import Data.Array( Array, assocs )

    import GHC.Utils.Misc
    import GHC.Utils.Outputable
    import GHC.Utils.Panic

    {-
    ************************************************************************
    *                                                                      *
              Type checked family instance heads
    *                                                                      *
    ************************************************************************

    Note [FamInsts and CoAxioms]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    * CoAxioms and FamInsts are just like
      DFunIds  and ClsInsts

    * A CoAxiom is a System-FC thing: it can relate any two types

    * A FamInst is a Haskell source-language thing, corresponding
      to a type/data family instance declaration.
        - The FamInst contains a CoAxiom, which is the evidence
          for the instance

        - The LHS of the CoAxiom is always of form F ty1 .. tyn
          where F is a type family
    -}

    data FamInst  -- See Note [FamInsts and CoAxioms]
      = FamInst { fi_axiom  :: CoAxiom Unbranched -- The new coercion axiom
                                                  -- introduced by this family
                                                  -- instance
                     -- INVARIANT: apart from freshening (see below)
                     --    fi_tvs = cab_tvs of the (single) axiom branch
                     --    fi_cvs = cab_cvs ...ditto...
                     --    fi_tys = cab_lhs ...ditto...
                     --    fi_rhs = cab_rhs ...ditto...

                , fi_flavor :: FamFlavor

                -- Everything below here is a redundant,
                -- cached version of the two things above
                -- except that the TyVars are freshened
                , fi_fam   :: Name          -- Family name

                    -- Used for "rough matching"; same idea as for class instances
                    -- See Note [Rough matching in class and family instances]
                    -- in GHC.Core.Unify
                , fi_tcs   :: [RoughMatchTc]  -- Top of type args
                    -- INVARIANT: fi_tcs = roughMatchTcs fi_tys

                -- Used for "proper matching"; ditto
                , fi_tvs :: [TyVar]      -- Template tyvars for full match
                , fi_cvs :: [CoVar]      -- Template covars for full match
                     -- Like ClsInsts, these variables are always fresh
                     -- See Note [Template tyvars are fresh] in GHC.Core.InstEnv

                , fi_tys    :: [Type]       --   The LHS type patterns
                -- May be eta-reduced; see Note [Eta reduction for data families]
                -- in GHC.Core.Coercion.Axiom

                , fi_rhs :: Type         --   the RHS, with its freshened vars
                }

    data FamFlavor
      = SynFamilyInst         -- A synonym family
      | DataFamilyInst TyCon  -- A data family, with its representation TyCon

    {-
    Note [Arity of data families]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Data family instances might legitimately be over- or under-saturated.

    Under-saturation has two potential causes:
     U1) Eta reduction. See Note [Eta reduction for data families] in
         GHC.Core.Coercion.Axiom.
     U2) When the user has specified a return kind instead of written out patterns.
         Example:

           data family Sing (a :: k)
           data instance Sing :: Bool -> Type

         The data family tycon Sing has an arity of 2, the k and the a. But
         the data instance has only one pattern, Bool (standing in for k).
         This instance is equivalent to `data instance Sing (a :: Bool)`, but
         without the last pattern, we have an under-saturated data family instance.
         On its own, this example is not compelling enough to add support for
         under-saturation, but U1 makes this feature more compelling.

    Over-saturation is also possible:
      O1) If the data family's return kind is a type variable (see also #12369),
          an instance might legitimately have more arguments than the family.
          Example:

            data family Fix :: (Type -> k) -> k
            data instance Fix f = MkFix1 (f (Fix f))
            data instance Fix f x = MkFix2 (f (Fix f x) x)

          In the first instance here, the k in the data family kind is chosen to
          be Type. In the second, it's (Type -> Type).

          However, we require that any over-saturation is eta-reducible. That is,
          we require that any extra patterns be bare unrepeated type variables;
          see Note [Eta reduction for data families] in GHC.Core.Coercion.Axiom.
          Accordingly, the FamInst is never over-saturated.

    Why can we allow such flexibility for data families but not for type families?
    Because data families can be decomposed -- that is, they are generative and
    injective. A Type family is neither and so always must be applied to all its
    arguments.
    -}

    -- Obtain the axiom of a family instance
    famInstAxiom :: FamInst -> CoAxiom Unbranched
    famInstAxiom = fi_axiom

    -- Split the left-hand side of the FamInst
    famInstSplitLHS :: FamInst -> (TyCon, [Type])
    famInstSplitLHS (FamInst { fi_axiom = axiom, fi_tys = lhs })
      = (coAxiomTyCon axiom, lhs)

    -- Get the RHS of the FamInst
    famInstRHS :: FamInst -> Type
    famInstRHS = fi_rhs

    -- Get the family TyCon of the FamInst
    famInstTyCon :: FamInst -> TyCon
    famInstTyCon = coAxiomTyCon . famInstAxiom

    -- Return the representation TyCons introduced by data family instances, if any
    famInstsRepTyCons :: [FamInst] -> [TyCon]
    famInstsRepTyCons fis = [tc | FamInst { fi_flavor = DataFamilyInst tc } <- fis]

    -- Extracts the TyCon for this *data* (or newtype) instance
    famInstRepTyCon_maybe :: FamInst -> Maybe TyCon
    famInstRepTyCon_maybe fi
      = case fi_flavor fi of
           DataFamilyInst tycon -> Just tycon
           SynFamilyInst        -> Nothing

    dataFamInstRepTyCon :: FamInst -> TyCon
    dataFamInstRepTyCon fi
      = case fi_flavor fi of
           DataFamilyInst tycon -> tycon
           SynFamilyInst        -> pprPanic "dataFamInstRepTyCon" (ppr fi)

    {-
    ************************************************************************
    *                                                                      *
            Pretty printing
    *                                                                      *
    ************************************************************************
    -}

    instance NamedThing FamInst where
       getName = coAxiomName . fi_axiom

    instance Outputable FamInst where
       ppr = pprFamInst

    pprFamInst :: FamInst -> SDoc
    -- Prints the FamInst as a family instance declaration
    -- NB: This function, FamInstEnv.pprFamInst, is used only for internal,
    --     debug printing. See GHC.Types.TyThing.Ppr.pprFamInst for printing for the user
    pprFamInst (FamInst { fi_flavor = flavor, fi_axiom = ax
                        , fi_tvs = tvs, fi_tys = tys, fi_rhs = rhs })
      = hang (ppr_tc_sort <+> text "instance"
                 <+> pprCoAxBranchUser (coAxiomTyCon ax) (coAxiomSingleBranch ax))
           2 (whenPprDebug debug_stuff)
      where
        ppr_tc_sort = case flavor of
                         SynFamilyInst             -> text "type"
                         DataFamilyInst tycon
                           | isDataTyCon     tycon -> text "data"
                           | isNewTyCon      tycon -> text "newtype"
                           | isAbstractTyCon tycon -> text "data"
                           | otherwise             -> text "WEIRD" <+> ppr tycon

        debug_stuff = vcat [ text "Coercion axiom:" <+> ppr ax
                           , text "Tvs:" <+> ppr tvs
                           , text "LHS:" <+> ppr tys
                           , text "RHS:" <+> ppr rhs ]

    pprFamInsts :: [FamInst] -> SDoc
    pprFamInsts finsts = vcat (map pprFamInst finsts)

    {-
    Note [Lazy axiom match]
    ~~~~~~~~~~~~~~~~~~~~~~~
    It is Vitally Important that mkImportedFamInst is *lazy* in its axiom
    parameter. The axiom is loaded lazily, via a forkM, in GHC.IfaceToCore. Sometime
    later, mkImportedFamInst is called using that axiom. However, the axiom
    may itself depend on entities which are not yet loaded as of the time
    of the mkImportedFamInst. Thus, if mkImportedFamInst eagerly looks at the
    axiom, a dependency loop spontaneously appears and GHC hangs. The solution
    is simply for mkImportedFamInst never, ever to look inside of the axiom
    until everything else is good and ready to do so. We can assume that this
    readiness has been achieved when some other code pulls on the axiom in the
    FamInst. Thus, we pattern match on the axiom lazily (in the where clause,
    not in the parameter list) and we assert the consistency of names there
    also.
    -}

    -- Make a family instance representation from the information found in an
    -- interface file.  In particular, we get the rough match info from the iface
    -- (instead of computing it here).
    mkImportedFamInst :: Name               -- Name of the family
                      -> [RoughMatchTc]     -- Rough match info
                      -> CoAxiom Unbranched -- Axiom introduced
                      -> FamInst            -- Resulting family instance
    mkImportedFamInst fam mb_tcs axiom
      = FamInst {
          fi_fam    = fam,
          fi_tcs    = mb_tcs,
          fi_tvs    = tvs,
          fi_cvs    = cvs,
          fi_tys    = tys,
          fi_rhs    = rhs,
          fi_axiom  = axiom,
          fi_flavor = flavor }
      where
         -- See Note [Lazy axiom match]
         ~(CoAxBranch { cab_lhs = tys
                      , cab_tvs = tvs
                      , cab_cvs = cvs
                      , cab_rhs = rhs }) = coAxiomSingleBranch axiom

             -- Derive the flavor for an imported FamInst rather disgustingly
             -- Maybe we should store it in the IfaceFamInst?
         flavor = case splitTyConApp_maybe rhs of
                    Just (tc, _)
                      | Just ax' <- tyConFamilyCoercion_maybe tc
                      , ax' == axiom
                      -> DataFamilyInst tc
                    _ -> SynFamilyInst

    {-
    ************************************************************************
    *                                                                      *
                    FamInstEnv
    *                                                                      *
    ************************************************************************

    Note [FamInstEnv]
    ~~~~~~~~~~~~~~~~~
    A FamInstEnv maps a family name to the list of known instances for that family.

    The same FamInstEnv includes both 'data family' and 'type family' instances.
    Type families are reduced during type inference, but not data families;
    the user explains when to use a data family instance by using constructors
    and pattern matching.

    Nevertheless it is still useful to have data families in the FamInstEnv:

     - For finding overlaps and conflicts

     - For finding the representation type...see FamInstEnv.topNormaliseType
       and its call site in GHC.Core.Opt.Simplify

     - In standalone deriving instance Eq (T [Int]) we need to find the
       representation type for T [Int]

    Note [Varying number of patterns for data family axioms]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    For data families, the number of patterns may vary between instances.
    For example
       data family T a b
       data instance T Int a = T1 a | T2
       data instance T Bool [a] = T3 a

    Then we get a data type for each instance, and an axiom:
       data TInt a = T1 a | T2
       data TBoolList a = T3 a

       axiom ax7   :: T Int ~ TInt   -- Eta-reduced
       axiom ax8 a :: T Bool [a] ~ TBoolList a

    These two axioms for T, one with one pattern, one with two;
    see Note [Eta reduction for data families] in GHC.Core.Coercion.Axiom

    Note [FamInstEnv determinism]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We turn FamInstEnvs into a list in some places that don't directly affect
    the ABI. That happens in family consistency checks and when producing output
    for `:info`. Unfortunately that nondeterminism is nonlocal and it's hard
    to tell what it affects without following a chain of functions. It's also
    easy to accidentally make that nondeterminism affect the ABI. Furthermore
    the envs should be relatively small, so it should be free to use deterministic
    maps here. Testing with nofib and validate detected no difference between
    UniqFM and UniqDFM.
    See Note [Deterministic UniqFM].
    -}

    -- Internally we sometimes index by Name instead of TyCon despite
    -- of what the type says. This is safe since
    -- getUnique (tyCon) == getUniqe (tcName tyCon)
    type FamInstEnv = UniqDFM TyCon FamilyInstEnv  -- Maps a family to its instances
         -- See Note [FamInstEnv]
         -- See Note [FamInstEnv determinism]

    type FamInstEnvs = (FamInstEnv, FamInstEnv)
         -- External package inst-env, Home-package inst-env

    newtype FamilyInstEnv
      = FamIE [FamInst]     -- The instances for a particular family, in any order

    instance Outputable FamilyInstEnv where
      ppr (FamIE fs) = text "FamIE" <+> vcat (map ppr fs)

    -- | Index a FamInstEnv by the tyCons name.
    toNameInstEnv :: FamInstEnv -> UniqDFM Name FamilyInstEnv
    toNameInstEnv = unsafeCastUDFMKey

    -- | Create a FamInstEnv from Name indices.
    fromNameInstEnv :: UniqDFM Name FamilyInstEnv -> FamInstEnv
    fromNameInstEnv = unsafeCastUDFMKey

    -- INVARIANTS:
    --  * The fs_tvs are distinct in each FamInst
    --      of a range value of the map (so we can safely unify them)

    emptyFamInstEnvs :: (FamInstEnv, FamInstEnv)
    emptyFamInstEnvs = (emptyFamInstEnv, emptyFamInstEnv)

    emptyFamInstEnv :: FamInstEnv
    emptyFamInstEnv = emptyUDFM

    famInstEnvElts :: FamInstEnv -> [FamInst]
    famInstEnvElts fi = [elt | FamIE elts <- eltsUDFM fi, elt <- elts]
      -- See Note [FamInstEnv determinism]

    famInstEnvSize :: FamInstEnv -> Int
    famInstEnvSize = nonDetStrictFoldUDFM (\(FamIE elt) sum -> sum + length elt) 0
      -- It's OK to use nonDetStrictFoldUDFM here since we're just computing the
      -- size.

    familyInstances :: (FamInstEnv, FamInstEnv) -> TyCon -> [FamInst]
    familyInstances (pkg_fie, home_fie) fam
      = get home_fie ++ get pkg_fie
      where
        get env = case lookupUDFM env fam of
                    Just (FamIE insts) -> insts
                    Nothing                      -> []

    extendFamInstEnvList :: FamInstEnv -> [FamInst] -> FamInstEnv
    extendFamInstEnvList inst_env fis = foldl' extendFamInstEnv inst_env fis

    extendFamInstEnv :: FamInstEnv -> FamInst -> FamInstEnv
    extendFamInstEnv inst_env
                     ins_item@(FamInst {fi_fam = cls_nm})
      = fromNameInstEnv $ addToUDFM_C add (toNameInstEnv inst_env) cls_nm (FamIE [ins_item])
      where
        add (FamIE items) _ = FamIE (ins_item:items)

    {-
    ************************************************************************
    *                                                                      *
                    Compatibility
    *                                                                      *
    ************************************************************************

    Note [Apartness]
    ~~~~~~~~~~~~~~~~
    In dealing with closed type families, we must be able to check that one type
    will never reduce to another. This check is called /apartness/. The check
    is always between a target (which may be an arbitrary type) and a pattern.
    Here is how we do it:

    apart(target, pattern) = not (unify(flatten(target), pattern))

    where flatten (implemented in flattenTys, below) converts all type-family
    applications into fresh variables. (See
    Note [Flattening type-family applications when matching instances] in GHC.Core.Unify.)

    Note [Compatibility]
    ~~~~~~~~~~~~~~~~~~~~
    Two patterns are /compatible/ if either of the following conditions hold:
    1) The patterns are apart.
    2) The patterns unify with a substitution S, and their right hand sides
    equal under that substitution.

    For open type families, only compatible instances are allowed. For closed
    type families, the story is slightly more complicated. Consider the following:

    type family F a where
      F Int = Bool
      F a   = Int

    g :: Show a => a -> F a
    g x = length (show x)

    Should that type-check? No. We need to allow for the possibility that 'a'
    might be Int and therefore 'F a' should be Bool. We can simplify 'F a' to Int
    only when we can be sure that 'a' is not Int.

    To achieve this, after finding a possible match within the equations, we have to
    go back to all previous equations and check that, under the
    substitution induced by the match, other branches are surely apart. (See
    Note [Apartness].) This is similar to what happens with class
    instance selection, when we need to guarantee that there is only a match and
    no unifiers. The exact algorithm is different here because the
    potentially-overlapping group is closed.

    As another example, consider this:

    type family G x where
      G Int = Bool
      G a   = Double

    type family H y
    -- no instances

    Now, we want to simplify (G (H Char)). We can't, because (H Char) might later
    simplify to be Int. So, (G (H Char)) is stuck, for now.

    While everything above is quite sound, it isn't as expressive as we'd like.
    Consider this:

    type family J a where
      J Int = Int
      J a   = a

    Can we simplify (J b) to b? Sure we can. Yes, the first equation matches if
    b is instantiated with Int, but the RHSs coincide there, so it's all OK.

    So, the rule is this: when looking up a branch in a closed type family, we
    find a branch that matches the target, but then we make sure that the target
    is apart from every previous *incompatible* branch. We don't check the
    branches that are compatible with the matching branch, because they are either
    irrelevant (clause 1 of compatible) or benign (clause 2 of compatible).

    Note [Compatibility of eta-reduced axioms]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    In newtype instances of data families we eta-reduce the axioms,
    See Note [Eta reduction for data families] in GHC.Core.Coercion.Axiom. This means that
    we sometimes need to test compatibility of two axioms that were eta-reduced to
    different degrees, e.g.:

    data family D a b c
    newtype instance D a Int c = DInt (Maybe a)
      -- D a Int ~ Maybe
      -- lhs = [a, Int]
    newtype instance D Bool Int Char = DIntChar Float
      -- D Bool Int Char ~ Float
      -- lhs = [Bool, Int, Char]

    These are obviously incompatible. We could detect this by saturating
    (eta-expanding) the shorter LHS with fresh tyvars until the lists are of
    equal length, but instead we can just remove the tail of the longer list, as
    those types will simply unify with the freshly introduced tyvars.

    By doing this, in case the LHS are unifiable, the yielded substitution won't
    mention the tyvars that appear in the tail we dropped off, and we might try
    to test equality RHSes of different kinds, but that's fine since this case
    occurs only for data families, where the RHS is a unique tycon and the equality
    fails anyway.
    -}

    -- See Note [Compatibility]
    compatibleBranches :: CoAxBranch -> CoAxBranch -> Bool
    compatibleBranches (CoAxBranch { cab_lhs = lhs1, cab_rhs = rhs1 })
                       (CoAxBranch { cab_lhs = lhs2, cab_rhs = rhs2 })
      = let (commonlhs1, commonlhs2) = zipAndUnzip lhs1 lhs2
                 -- See Note [Compatibility of eta-reduced axioms]
        in case tcUnifyTysFG alwaysBindFun commonlhs1 commonlhs2 of
          SurelyApart -> True
          Unifiable subst
            | Type.substTyAddInScope subst rhs1 `eqType`
              Type.substTyAddInScope subst rhs2
            -> True
          _ -> False

    -- | Result of testing two type family equations for injectiviy.
    data InjectivityCheckResult
       = InjectivityAccepted
        -- ^ Either RHSs are distinct or unification of RHSs leads to unification of
        -- LHSs
       | InjectivityUnified CoAxBranch CoAxBranch
        -- ^ RHSs unify but LHSs don't unify under that substitution.  Relevant for
        -- closed type families where equation after unification might be
        -- overlpapped (in which case it is OK if they don't unify).  Constructor
        -- stores axioms after unification.

    -- | Check whether two type family axioms don't violate injectivity annotation.
    injectiveBranches :: [Bool] -> CoAxBranch -> CoAxBranch
                      -> InjectivityCheckResult
    injectiveBranches injectivity
                      ax1@(CoAxBranch { cab_lhs = lhs1, cab_rhs = rhs1 })
                      ax2@(CoAxBranch { cab_lhs = lhs2, cab_rhs = rhs2 })
      -- See Note [Verifying injectivity annotation], case 1.
      = let getInjArgs  = filterByList injectivity
        in case tcUnifyTyWithTFs True rhs1 rhs2 of -- True = two-way pre-unification
           Nothing -> InjectivityAccepted
             -- RHS are different, so equations are injective.
             -- This is case 1A from Note [Verifying injectivity annotation]
           Just subst -> -- RHS unify under a substitution
            let lhs1Subst = Type.substTys subst (getInjArgs lhs1)
                lhs2Subst = Type.substTys subst (getInjArgs lhs2)
            -- If LHSs are equal under the substitution used for RHSs then this pair
            -- of equations does not violate injectivity annotation. If LHSs are not
            -- equal under that substitution then this pair of equations violates
            -- injectivity annotation, but for closed type families it still might
            -- be the case that one LHS after substitution is unreachable.
            in if eqTypes lhs1Subst lhs2Subst  -- check case 1B1 from Note.
               then InjectivityAccepted
               else InjectivityUnified ( ax1 { cab_lhs = Type.substTys subst lhs1
                                             , cab_rhs = Type.substTy  subst rhs1 })
                                       ( ax2 { cab_lhs = Type.substTys subst lhs2
                                             , cab_rhs = Type.substTy  subst rhs2 })
                    -- payload of InjectivityUnified used only for check 1B2, only
                    -- for closed type families

    -- takes a CoAxiom with unknown branch incompatibilities and computes
    -- the compatibilities
    -- See Note [Storing compatibility] in GHC.Core.Coercion.Axiom
    computeAxiomIncomps :: [CoAxBranch] -> [CoAxBranch]
    computeAxiomIncomps branches
      = snd (mapAccumL go [] branches)
      where
        go :: [CoAxBranch] -> CoAxBranch -> ([CoAxBranch], CoAxBranch)
        go prev_brs cur_br
           = (cur_br : prev_brs, new_br)
           where
             new_br = cur_br { cab_incomps = mk_incomps prev_brs cur_br }

        mk_incomps :: [CoAxBranch] -> CoAxBranch -> [CoAxBranch]
        mk_incomps prev_brs cur_br
           = filter (not . compatibleBranches cur_br) prev_brs

    {-
    ************************************************************************
    *                                                                      *
               Constructing axioms
        These functions are here because tidyType / tcUnifyTysFG
        are not available in GHC.Core.Coercion.Axiom

        Also computeAxiomIncomps is too sophisticated for CoAxiom
    *                                                                      *
    ************************************************************************

    Note [Tidy axioms when we build them]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Like types and classes, we build axioms fully quantified over all
    their variables, and tidy them when we build them. For example,
    we print out axioms and don't want to print stuff like
        F k k a b = ...
    Instead we must tidy those kind variables.  See #7524.

    We could instead tidy when we print, but that makes it harder to get
    things like injectivity errors to come out right. Danger of
         Type family equation violates injectivity annotation.
         Kind variable ‘k’ cannot be inferred from the right-hand side.
         In the type family equation:
            PolyKindVars @[k1] @[k2] ('[] @k1) = '[] @k2

    Note [Always number wildcard types in CoAxBranch]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider the following example (from the DataFamilyInstanceLHS test case):

      data family Sing (a :: k)
      data instance Sing (_ :: MyKind) where
          SingA :: Sing A
          SingB :: Sing B

    If we're not careful during tidying, then when this program is compiled with
    -ddump-types, we'll get the following information:

      COERCION AXIOMS
        axiom DataFamilyInstanceLHS.D:R:SingMyKind_0 ::
          Sing _ = DataFamilyInstanceLHS.R:SingMyKind_ _

    It's misleading to have a wildcard type appearing on the RHS like
    that. To avoid this issue, when building a CoAxiom (which is what eventually
    gets printed above), we tidy all the variables in an env that already contains
    '_'. Thus, any variable named '_' will be renamed, giving us the nicer output
    here:

      COERCION AXIOMS
        axiom DataFamilyInstanceLHS.D:R:SingMyKind_0 ::
          Sing _1 = DataFamilyInstanceLHS.R:SingMyKind_ _1

    Which is at least legal syntax.

    See also Note [CoAxBranch type variables] in GHC.Core.Coercion.Axiom; note that we
    are tidying (changing OccNames only), not freshening, in accordance with
    that Note.
    -}

    -- all axiom roles are Nominal, as this is only used with type families
    mkCoAxBranch :: [TyVar] -- original, possibly stale, tyvars
                 -> [TyVar] -- Extra eta tyvars
                 -> [CoVar] -- possibly stale covars
                 -> [Type]  -- LHS patterns
                 -> Type    -- RHS
                 -> [Role]
                 -> SrcSpan
                 -> CoAxBranch
    mkCoAxBranch tvs eta_tvs cvs lhs rhs roles loc
      = CoAxBranch { cab_tvs     = tvs'
                   , cab_eta_tvs = eta_tvs'
                   , cab_cvs     = cvs'
                   , cab_lhs     = tidyTypes env lhs
                   , cab_roles   = roles
                   , cab_rhs     = tidyType env rhs
                   , cab_loc     = loc
                   , cab_incomps = placeHolderIncomps }
      where
        (env1, tvs')     = tidyVarBndrs init_tidy_env tvs
        (env2, eta_tvs') = tidyVarBndrs env1          eta_tvs
        (env,  cvs')     = tidyVarBndrs env2          cvs
        -- See Note [Tidy axioms when we build them]
        -- See also Note [CoAxBranch type variables] in GHC.Core.Coercion.Axiom

        init_occ_env = initTidyOccEnv [mkTyVarOcc "_"]
        init_tidy_env = mkEmptyTidyEnv init_occ_env
        -- See Note [Always number wildcard types in CoAxBranch]

    -- all of the following code is here to avoid mutual dependencies with
    -- Coercion
    mkBranchedCoAxiom :: Name -> TyCon -> [CoAxBranch] -> CoAxiom Branched
    mkBranchedCoAxiom ax_name fam_tc branches
      = CoAxiom { co_ax_unique   = nameUnique ax_name
                , co_ax_name     = ax_name
                , co_ax_tc       = fam_tc
                , co_ax_role     = Nominal
                , co_ax_implicit = False
                , co_ax_branches = manyBranches (computeAxiomIncomps branches) }

    mkUnbranchedCoAxiom :: Name -> TyCon -> CoAxBranch -> CoAxiom Unbranched
    mkUnbranchedCoAxiom ax_name fam_tc branch
      = CoAxiom { co_ax_unique   = nameUnique ax_name
                , co_ax_name     = ax_name
                , co_ax_tc       = fam_tc
                , co_ax_role     = Nominal
                , co_ax_implicit = False
                , co_ax_branches = unbranched (branch { cab_incomps = [] }) }

    mkSingleCoAxiom :: Role -> Name
                    -> [TyVar] -> [TyVar] -> [CoVar]
                    -> TyCon -> [Type] -> Type
                    -> CoAxiom Unbranched
    -- Make a single-branch CoAxiom, including making the branch itself
    -- Used for both type family (Nominal) and data family (Representational)
    -- axioms, hence passing in the Role
    mkSingleCoAxiom role ax_name tvs eta_tvs cvs fam_tc lhs_tys rhs_ty
      = CoAxiom { co_ax_unique   = nameUnique ax_name
                , co_ax_name     = ax_name
                , co_ax_tc       = fam_tc
                , co_ax_role     = role
                , co_ax_implicit = False
                , co_ax_branches = unbranched (branch { cab_incomps = [] }) }
      where
        branch = mkCoAxBranch tvs eta_tvs cvs lhs_tys rhs_ty
                              (map (const Nominal) tvs)
                              (getSrcSpan ax_name)

    -- | Create a coercion constructor (axiom) suitable for the given
    --   newtype 'TyCon'. The 'Name' should be that of a new coercion
    --   'CoAxiom', the 'TyVar's the arguments expected by the @newtype@ and
    --   the type the appropriate right hand side of the @newtype@, with
    --   the free variables a subset of those 'TyVar's.
    mkNewTypeCoAxiom :: Name -> TyCon -> [TyVar] -> [Role] -> Type -> CoAxiom Unbranched
    mkNewTypeCoAxiom name tycon tvs roles rhs_ty
      = CoAxiom { co_ax_unique   = nameUnique name
                , co_ax_name     = name
                , co_ax_implicit = True  -- See Note [Implicit axioms] in GHC.Core.TyCon
                , co_ax_role     = Representational
                , co_ax_tc       = tycon
                , co_ax_branches = unbranched (branch { cab_incomps = [] }) }
      where
        branch = mkCoAxBranch tvs [] [] (mkTyVarTys tvs) rhs_ty
                              roles (getSrcSpan name)

    {-
    ************************************************************************
    *                                                                      *
                    Looking up a family instance
    *                                                                      *
    ************************************************************************

    @lookupFamInstEnv@ looks up in a @FamInstEnv@, using a one-way match.
    Multiple matches are only possible in case of type families (not data
    families), and then, it doesn't matter which match we choose (as the
    instances are guaranteed confluent).

    We return the matching family instances and the type instance at which it
    matches.  For example, if we lookup 'T [Int]' and have a family instance

      data instance T [a] = ..

    desugared to

      data :R42T a = ..
      coe :Co:R42T a :: T [a] ~ :R42T a

    we return the matching instance '(FamInst{.., fi_tycon = :R42T}, Int)'.
    -}

    -- when matching a type family application, we get a FamInst,
    -- and the list of types the axiom should be applied to
    data FamInstMatch = FamInstMatch { fim_instance :: FamInst
                                     , fim_tys      :: [Type]
                                     , fim_cos      :: [Coercion]
                                     }
      -- See Note [Over-saturated matches]

    instance Outputable FamInstMatch where
      ppr (FamInstMatch { fim_instance = inst
                        , fim_tys      = tys
                        , fim_cos      = cos })
        = text "match with" <+> parens (ppr inst) <+> ppr tys <+> ppr cos

    lookupFamInstEnvByTyCon :: FamInstEnvs -> TyCon -> [FamInst]
    lookupFamInstEnvByTyCon (pkg_ie, home_ie) fam_tc
      = get pkg_ie ++ get home_ie
      where
        get ie = case lookupUDFM ie fam_tc of
                   Nothing          -> []
                   Just (FamIE fis) -> fis

    lookupFamInstEnv
        :: FamInstEnvs
        -> TyCon -> [Type]          -- What we are looking for
        -> [FamInstMatch]           -- Successful matches
    -- Precondition: the tycon is saturated (or over-saturated)

    lookupFamInstEnv
       = lookup_fam_inst_env match
       where
         match _ _ tpl_tys tys = tcMatchTys tpl_tys tys

    lookupFamInstEnvConflicts
        :: FamInstEnvs
        -> FamInst          -- Putative new instance
        -> [FamInstMatch]   -- Conflicting matches (don't look at the fim_tys field)
    -- E.g. when we are about to add
    --    f : type instance F [a] = a->a
    -- we do (lookupFamInstConflicts f [b])
    -- to find conflicting matches
    --
    -- Precondition: the tycon is saturated (or over-saturated)

    lookupFamInstEnvConflicts envs fam_inst@(FamInst { fi_axiom = new_axiom })
      = lookup_fam_inst_env my_unify envs fam tys
      where
        (fam, tys) = famInstSplitLHS fam_inst
            -- In example above,   fam tys' = F [b]

        my_unify (FamInst { fi_axiom = old_axiom }) tpl_tvs tpl_tys _
           = ASSERT2( tyCoVarsOfTypes tys `disjointVarSet` tpl_tvs,
                      (ppr fam <+> ppr tys) $$
                      (ppr tpl_tvs <+> ppr tpl_tys) )
                    -- Unification will break badly if the variables overlap
                    -- They shouldn't because we allocate separate uniques for them
             if compatibleBranches (coAxiomSingleBranch old_axiom) new_branch
               then Nothing
               else Just noSubst
          -- Note [Family instance overlap conflicts]

        noSubst = panic "lookupFamInstEnvConflicts noSubst"
        new_branch = coAxiomSingleBranch new_axiom

    --------------------------------------------------------------------------------
    --                 Type family injectivity checking bits                      --
    --------------------------------------------------------------------------------

    {- Note [Verifying injectivity annotation]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Injectivity means that the RHS of a type family uniquely determines the LHS (see
    Note [Type inference for type families with injectivity]).  The user informs us about
    injectivity using an injectivity annotation and it is GHC's task to verify that
    this annotation is correct w.r.t. type family equations. Whenever we see a new
    equation of a type family we need to make sure that adding this equation to the
    already known equations of a type family does not violate the injectivity annotation
    supplied by the user (see Note [Injectivity annotation]).  Of course if the type
    family has no injectivity annotation then no check is required.  But if a type
    family has injectivity annotation we need to make sure that the following
    conditions hold:

    1. For each pair of *different* equations of a type family, one of the following
       conditions holds:

       A:  RHSs are different. (Check done in GHC.Core.FamInstEnv.injectiveBranches)

       B1: OPEN TYPE FAMILIES: If the RHSs can be unified under some substitution
           then it must be possible to unify the LHSs under the same substitution.
           Example:

              type family FunnyId a = r | r -> a
              type instance FunnyId Int = Int
              type instance FunnyId a = a

           RHSs of these two equations unify under [ a |-> Int ] substitution.
           Under this substitution LHSs are equal therefore these equations don't
           violate injectivity annotation. (Check done in GHC.Core.FamInstEnv.injectiveBranches)

       B2: CLOSED TYPE FAMILIES: If the RHSs can be unified under some
           substitution then either the LHSs unify under the same substitution or
           the LHS of the latter equation is overlapped by earlier equations.
           Example 1:

              type family SwapIntChar a = r | r -> a where
                  SwapIntChar Int  = Char
                  SwapIntChar Char = Int
                  SwapIntChar a    = a

           Say we are checking the last two equations. RHSs unify under [ a |->
           Int ] substitution but LHSs don't. So we apply the substitution to LHS
           of last equation and check whether it is overlapped by any of previous
           equations. Since it is overlapped by the first equation we conclude
           that pair of last two equations does not violate injectivity
           annotation. (Check done in GHC.Tc.Validity.checkValidCoAxiom#gather_conflicts)

       A special case of B is when RHSs unify with an empty substitution ie. they
       are identical.

       If any of the above two conditions holds we conclude that the pair of
       equations does not violate injectivity annotation. But if we find a pair
       of equations where neither of the above holds we report that this pair
       violates injectivity annotation because for a given RHS we don't have a
       unique LHS. (Note that (B) actually implies (A).)

       Note that we only take into account these LHS patterns that were declared
       as injective.

    2. If an RHS of a type family equation is a bare type variable then
       all LHS variables (including implicit kind variables) also have to be bare.
       In other words, this has to be a sole equation of that type family and it has
       to cover all possible patterns.  So for example this definition will be
       rejected:

          type family W1 a = r | r -> a
          type instance W1 [a] = a

       If it were accepted we could call `W1 [W1 Int]`, which would reduce to
       `W1 Int` and then by injectivity we could conclude that `[W1 Int] ~ Int`,
       which is bogus. Checked FamInst.bareTvInRHSViolated.

    3. If the RHS of a type family equation is a type family application then the type
       family is rejected as not injective. This is checked by FamInst.isTFHeaded.

    4. If a LHS type variable that is declared as injective is not mentioned in an
       injective position in the RHS then the type family is rejected as not
       injective.  "Injective position" means either an argument to a type
       constructor or argument to a type family on injective position.
       There are subtleties here. See Note [Coverage condition for injective type families]
       in GHC.Tc.Instance.Family.

    Check (1) must be done for all family instances (transitively) imported. Other
    checks (2-4) should be done just for locally written equations, as they are checks
    involving just a single equation, not about interactions. Doing the other checks for
    imported equations led to #17405, as the behavior of check (4) depends on
    -XUndecidableInstances (see Note [Coverage condition for injective type families] in
    FamInst), which may vary between modules.

    See also Note [Injective type families] in GHC.Core.TyCon
    -}

    -- | Check whether an open type family equation can be added to already existing
    -- instance environment without causing conflicts with supplied injectivity
    -- annotations.  Returns list of conflicting axioms (type instance
    -- declarations).
    lookupFamInstEnvInjectivityConflicts
        :: [Bool]         -- injectivity annotation for this type family instance
                          -- INVARIANT: list contains at least one True value
        ->  FamInstEnvs   -- all type instances seens so far
        ->  FamInst       -- new type instance that we're checking
        -> [CoAxBranch]   -- conflicting instance declarations
    lookupFamInstEnvInjectivityConflicts injList (pkg_ie, home_ie)
                                 fam_inst@(FamInst { fi_axiom = new_axiom })
      -- See Note [Verifying injectivity annotation]. This function implements
      -- check (1.B1) for open type families described there.
      = lookup_inj_fam_conflicts home_ie ++ lookup_inj_fam_conflicts pkg_ie
        where
          fam        = famInstTyCon fam_inst
          new_branch = coAxiomSingleBranch new_axiom

          -- filtering function used by `lookup_inj_fam_conflicts` to check whether
          -- a pair of equations conflicts with the injectivity annotation.
          isInjConflict (FamInst { fi_axiom = old_axiom })
              | InjectivityAccepted <-
                injectiveBranches injList (coAxiomSingleBranch old_axiom) new_branch
              = False -- no conflict
              | otherwise = True

          lookup_inj_fam_conflicts ie
              | isOpenFamilyTyCon fam, Just (FamIE insts) <- lookupUDFM ie fam
              = map (coAxiomSingleBranch . fi_axiom) $
                filter isInjConflict insts
              | otherwise = []

    --------------------------------------------------------------------------------
    --                    Type family overlap checking bits                       --
    --------------------------------------------------------------------------------

    {-
    Note [Family instance overlap conflicts]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    - In the case of data family instances, any overlap is fundamentally a
      conflict (as these instances imply injective type mappings).

    - In the case of type family instances, overlap is admitted as long as
      the right-hand sides of the overlapping rules coincide under the
      overlap substitution.  eg
           type instance F a Int = a
           type instance F Int b = b
      These two overlap on (F Int Int) but then both RHSs are Int,
      so all is well. We require that they are syntactically equal;
      anything else would be difficult to test for at this stage.
    -}

    ------------------------------------------------------------
    -- Might be a one-way match or a unifier
    type MatchFun =  FamInst                -- The FamInst template
                  -> TyVarSet -> [Type]     --   fi_tvs, fi_tys of that FamInst
                  -> [Type]                 -- Target to match against
                  -> Maybe TCvSubst

    lookup_fam_inst_env'          -- The worker, local to this module
        :: MatchFun
        -> FamInstEnv
        -> TyCon -> [Type]        -- What we are looking for
        -> [FamInstMatch]
    lookup_fam_inst_env' match_fun ie fam match_tys
      | isOpenFamilyTyCon fam
      , Just (FamIE insts) <- lookupUDFM ie fam
      = find insts    -- The common case
      | otherwise = []
      where

        find [] = []
        find (item@(FamInst { fi_tcs = mb_tcs, fi_tvs = tpl_tvs, fi_cvs = tpl_cvs
                            , fi_tys = tpl_tys }) : rest)
            -- Fast check for no match, uses the "rough match" fields
          | instanceCantMatch rough_tcs mb_tcs
          = find rest

            -- Proper check
          | Just subst <- match_fun item (mkVarSet tpl_tvs) tpl_tys match_tys1
          = (FamInstMatch { fim_instance = item
                          , fim_tys      = substTyVars subst tpl_tvs `chkAppend` match_tys2
                          , fim_cos      = ASSERT( all (isJust . lookupCoVar subst) tpl_cvs )
                                           substCoVars subst tpl_cvs
                          })
            : find rest

            -- No match => try next
          | otherwise
          = find rest
          where
            (rough_tcs, match_tys1, match_tys2) = split_tys tpl_tys

          -- Precondition: the tycon is saturated (or over-saturated)

        -- Deal with over-saturation
        -- See Note [Over-saturated matches]
        split_tys tpl_tys
          | isTypeFamilyTyCon fam
          = pre_rough_split_tys

          | otherwise
          = let (match_tys1, match_tys2) = splitAtList tpl_tys match_tys
                rough_tcs = roughMatchTcs match_tys1
            in (rough_tcs, match_tys1, match_tys2)

        (pre_match_tys1, pre_match_tys2) = splitAt (tyConArity fam) match_tys
        pre_rough_split_tys
          = (roughMatchTcs pre_match_tys1, pre_match_tys1, pre_match_tys2)

    lookup_fam_inst_env           -- The worker, local to this module
        :: MatchFun
        -> FamInstEnvs
        -> TyCon -> [Type]        -- What we are looking for
        -> [FamInstMatch]         -- Successful matches

    -- Precondition: the tycon is saturated (or over-saturated)

    lookup_fam_inst_env match_fun (pkg_ie, home_ie) fam tys
      =  lookup_fam_inst_env' match_fun home_ie fam tys
      ++ lookup_fam_inst_env' match_fun pkg_ie  fam tys

    {-
    Note [Over-saturated matches]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    It's ok to look up an over-saturated type constructor.  E.g.
         type family F a :: * -> *
         type instance F (a,b) = Either (a->b)

    The type instance gives rise to a newtype TyCon (at a higher kind
    which you can't do in Haskell!):
         newtype FPair a b = FP (Either (a->b))

    Then looking up (F (Int,Bool) Char) will return a FamInstMatch
         (FPair, [Int,Bool,Char])
    The "extra" type argument [Char] just stays on the end.

    We handle data families and type families separately here:

     * For type families, all instances of a type family must have the
       same arity, so we can precompute the split between the match_tys
       and the overflow tys. This is done in pre_rough_split_tys.

     * For data family instances, though, we need to re-split for each
       instance, because the breakdown might be different for each
       instance.  Why?  Because of eta reduction; see
       Note [Eta reduction for data families] in GHC.Core.Coercion.Axiom.
    -}

    -- checks if one LHS is dominated by a list of other branches
    -- in other words, if an application would match the first LHS, it is guaranteed
    -- to match at least one of the others. The RHSs are ignored.
    -- This algorithm is conservative:
    --   True -> the LHS is definitely covered by the others
    --   False -> no information
    -- It is currently (Oct 2012) used only for generating errors for
    -- inaccessible branches. If these errors go unreported, no harm done.
    -- This is defined here to avoid a dependency from CoAxiom to Unify
    isDominatedBy :: CoAxBranch -> [CoAxBranch] -> Bool
    isDominatedBy branch branches
      = or $ map match branches
        where
          lhs = coAxBranchLHS branch
          match (CoAxBranch { cab_lhs = tys })
            = isJust $ tcMatchTys tys lhs

    {-
    ************************************************************************
    *                                                                      *
                    Choosing an axiom application
    *                                                                      *
    ************************************************************************

    The lookupFamInstEnv function does a nice job for *open* type families,
    but we also need to handle closed ones when normalising a type:
    -}

    reduceTyFamApp_maybe :: FamInstEnvs
                         -> Role              -- Desired role of result coercion
                         -> TyCon -> [Type]
                         -> Maybe (Coercion, Type)
    -- Attempt to do a *one-step* reduction of a type-family application
    --    but *not* newtypes
    -- Works on type-synonym families always; data-families only if
    --     the role we seek is representational
    -- It does *not* normalise the type arguments first, so this may not
    --     go as far as you want. If you want normalised type arguments,
    --     use topReduceTyFamApp_maybe
    --
    -- The TyCon can be oversaturated.
    -- Works on both open and closed families
    --
    -- Always returns a *homogeneous* coercion -- type family reductions are always
    -- homogeneous
    reduceTyFamApp_maybe envs role tc tys
      | Phantom <- role
      = Nothing

      | case role of
          Representational -> isOpenFamilyTyCon     tc
          _                -> isOpenTypeFamilyTyCon tc
           -- If we seek a representational coercion
           -- (e.g. the call in topNormaliseType_maybe) then we can
           -- unwrap data families as well as type-synonym families;
           -- otherwise only type-synonym families
      , FamInstMatch { fim_instance = FamInst { fi_axiom = ax }
                     , fim_tys      = inst_tys
                     , fim_cos      = inst_cos } : _ <- lookupFamInstEnv envs tc tys
          -- NB: Allow multiple matches because of compatible overlap

      = let co = mkUnbranchedAxInstCo role ax inst_tys inst_cos
            ty = coercionRKind co
        in Just (co, ty)

      | Just ax <- isClosedSynFamilyTyConWithAxiom_maybe tc
      , Just (ind, inst_tys, inst_cos) <- chooseBranch ax tys
      = let co = mkAxInstCo role ax ind inst_tys inst_cos
            ty = coercionRKind co
        in Just (co, ty)

      | Just ax           <- isBuiltInSynFamTyCon_maybe tc
      , Just (coax,ts,ty) <- sfMatchFam ax tys
      , role == coaxrRole coax
      = let co = mkAxiomRuleCo coax (zipWith mkReflCo (coaxrAsmpRoles coax) ts)
        in Just (co, ty)

      | otherwise
      = Nothing

    -- The axiom can be oversaturated. (Closed families only.)
    chooseBranch :: CoAxiom Branched -> [Type]
                 -> Maybe (BranchIndex, [Type], [Coercion])  -- found match, with args
    chooseBranch axiom tys
      = do { let num_pats = coAxiomNumPats axiom
                 (target_tys, extra_tys) = splitAt num_pats tys
                 branches = coAxiomBranches axiom
           ; (ind, inst_tys, inst_cos)
               <- findBranch (unMkBranches branches) target_tys
           ; return ( ind, inst_tys `chkAppend` extra_tys, inst_cos ) }

    -- The axiom must *not* be oversaturated
    findBranch :: Array BranchIndex CoAxBranch
               -> [Type]
               -> Maybe (BranchIndex, [Type], [Coercion])
        -- coercions relate requested types to returned axiom LHS at role N
    findBranch branches target_tys
      = foldr go Nothing (assocs branches)
      where
        go :: (BranchIndex, CoAxBranch)
           -> Maybe (BranchIndex, [Type], [Coercion])
           -> Maybe (BranchIndex, [Type], [Coercion])
        go (index, branch) other
          = let (CoAxBranch { cab_tvs = tpl_tvs, cab_cvs = tpl_cvs
                            , cab_lhs = tpl_lhs
                            , cab_incomps = incomps }) = branch
                in_scope = mkInScopeSet (unionVarSets $
                                map (tyCoVarsOfTypes . coAxBranchLHS) incomps)
                -- See Note [Flattening type-family applications when matching instances]
                -- in GHC.Core.Unify
                flattened_target = flattenTys in_scope target_tys
            in case tcMatchTys tpl_lhs target_tys of
            Just subst -- matching worked. now, check for apartness.
              |  apartnessCheck flattened_target branch
              -> -- matching worked & we're apart from all incompatible branches.
                 -- success
                 ASSERT( all (isJust . lookupCoVar subst) tpl_cvs )
                 Just (index, substTyVars subst tpl_tvs, substCoVars subst tpl_cvs)

            -- failure. keep looking
            _ -> other

    -- | Do an apartness check, as described in the "Closed Type Families" paper
    -- (POPL '14). This should be used when determining if an equation
    -- ('CoAxBranch') of a closed type family can be used to reduce a certain target
    -- type family application.
    apartnessCheck :: [Type]
      -- ^ /flattened/ target arguments. Make sure they're flattened! See
      -- Note [Flattening type-family applications when matching instances]
      -- in GHC.Core.Unify.
                   -> CoAxBranch -- ^ the candidate equation we wish to use
                                 -- Precondition: this matches the target
                   -> Bool       -- ^ True <=> equation can fire
    apartnessCheck flattened_target (CoAxBranch { cab_incomps = incomps })
      = all (isSurelyApart
             . tcUnifyTysFG alwaysBindFun flattened_target
             . coAxBranchLHS) incomps
      where
        isSurelyApart SurelyApart = True
        isSurelyApart _           = False

    {-
    ************************************************************************
    *                                                                      *
                    Looking up a family instance
    *                                                                      *
    ************************************************************************

    Note [Normalising types]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    The topNormaliseType function removes all occurrences of type families
    and newtypes from the top-level structure of a type. normaliseTcApp does
    the type family lookup and is fairly straightforward. normaliseType is
    a little more involved.

    The complication comes from the fact that a type family might be used in the
    kind of a variable bound in a forall. We wish to remove this type family
    application, but that means coming up with a fresh variable (with the new
    kind). Thus, we need a substitution to be built up as we recur through the
    type. However, an ordinary TCvSubst just won't do: when we hit a type variable
    whose kind has changed during normalisation, we need both the new type
    variable *and* the coercion. We could conjure up a new VarEnv with just this
    property, but a usable substitution environment already exists:
    LiftingContexts from the liftCoSubst family of functions, defined in GHC.Core.Coercion.
    A LiftingContext maps a type variable to a coercion and a coercion variable to
    a pair of coercions. Let's ignore coercion variables for now. Because the
    coercion a type variable maps to contains the destination type (via
    coercionKind), we don't need to store that destination type separately. Thus,
    a LiftingContext has what we need: a map from type variables to (Coercion,
    Type) pairs.

    We also benefit because we can piggyback on the liftCoSubstVarBndr function to
    deal with binders. However, I had to modify that function to work with this
    application. Thus, we now have liftCoSubstVarBndrUsing, which takes
    a function used to process the kind of the binder. We don't wish
    to lift the kind, but instead normalise it. So, we pass in a callback function
    that processes the kind of the binder.

    After that brilliant explanation of all this, I'm sure you've forgotten the
    dangling reference to coercion variables. What do we do with those? Nothing at
    all. The point of normalising types is to remove type family applications, but
    there's no sense in removing these from coercions. We would just get back a
    new coercion witnessing the equality between the same types as the original
    coercion. Because coercions are irrelevant anyway, there is no point in doing
    this. So, whenever we encounter a coercion, we just say that it won't change.
    That's what the CoercionTy case is doing within normalise_type.

    Note [Normalisation and type synonyms]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We need to be a bit careful about normalising in the presence of type
    synonyms (#13035).  Suppose S is a type synonym, and we have
       S t1 t2
    If S is family-free (on its RHS) we can just normalise t1 and t2 and
    reconstruct (S t1' t2').   Expanding S could not reveal any new redexes
    because type families are saturated.

    But if S has a type family on its RHS we expand /before/ normalising
    the args t1, t2.  If we normalise t1, t2 first, we'll re-normalise them
    after expansion, and that can lead to /exponential/ behaviour; see #13035.

    Notice, though, that expanding first can in principle duplicate t1,t2,
    which might contain redexes. I'm sure you could conjure up an exponential
    case by that route too, but it hasn't happened in practice yet!
    -}

    topNormaliseType :: FamInstEnvs -> Type -> Type
    topNormaliseType env ty = case topNormaliseType_maybe env ty of
                                Just (_co, ty') -> ty'
                                Nothing         -> ty

    topNormaliseType_maybe :: FamInstEnvs -> Type -> Maybe (Coercion, Type)

    -- ^ Get rid of *outermost* (or toplevel)
    --      * type function redex
    --      * data family redex
    --      * newtypes
    -- returning an appropriate Representational coercion.  Specifically, if
    --   topNormaliseType_maybe env ty = Just (co, ty')
    -- then
    --   (a) co :: ty ~R ty'
    --   (b) ty' is not a newtype, and is not a type-family or data-family redex
    --
    -- However, ty' can be something like (Maybe (F ty)), where
    -- (F ty) is a redex.
    --
    -- Always operates homogeneously: the returned type has the same kind as the
    -- original type, and the returned coercion is always homogeneous.
    topNormaliseType_maybe env ty
      = do { ((co, mkind_co), nty) <- topNormaliseTypeX stepper combine ty
           ; return $ case mkind_co of
               MRefl       -> (co, nty)
               MCo kind_co -> let nty_casted = nty `mkCastTy` mkSymCo kind_co
                                  final_co   = mkCoherenceRightCo Representational nty
                                                                  (mkSymCo kind_co) co
                              in (final_co, nty_casted) }
      where
        stepper = unwrapNewTypeStepper' `composeSteppers` tyFamStepper

        combine (c1, mc1) (c2, mc2) = (c1 `mkTransCo` c2, mc1 `mkTransMCo` mc2)

        unwrapNewTypeStepper' :: NormaliseStepper (Coercion, MCoercionN)
        unwrapNewTypeStepper' rec_nts tc tys
          = mapStepResult (, MRefl) $ unwrapNewTypeStepper rec_nts tc tys

          -- second coercion below is the kind coercion relating the original type's kind
          -- to the normalised type's kind
        tyFamStepper :: NormaliseStepper (Coercion, MCoercionN)
        tyFamStepper rec_nts tc tys  -- Try to step a type/data family
          = case topReduceTyFamApp_maybe env tc tys of
              Just (co, rhs, res_co) -> NS_Step rec_nts rhs (co, res_co)
              _                      -> NS_Done

    ---------------
    normaliseTcApp :: FamInstEnvs -> Role -> TyCon -> [Type] -> (Coercion, Type)
    -- See comments on normaliseType for the arguments of this function
    normaliseTcApp env role tc tys
      = initNormM env role (tyCoVarsOfTypes tys) $
        normalise_tc_app tc tys

    -- See Note [Normalising types] about the LiftingContext
    normalise_tc_app :: TyCon -> [Type] -> NormM (Coercion, Type)
    normalise_tc_app tc tys
      | Just (tenv, rhs, tys') <- expandSynTyCon_maybe tc tys
      , not (isFamFreeTyCon tc)  -- Expand and try again
      = -- A synonym with type families in the RHS
        -- Expand and try again
        -- See Note [Normalisation and type synonyms]
        normalise_type (mkAppTys (substTy (mkTvSubstPrs tenv) rhs) tys')

      | isFamilyTyCon tc
      = -- A type-family application
        do { env <- getEnv
           ; role <- getRole
           ; (args_co, ntys, res_co) <- normalise_tc_args tc tys
           ; case reduceTyFamApp_maybe env role tc ntys of
               Just (first_co, ty')
                 -> do { (rest_co,nty) <- normalise_type ty'
                       ; return (assemble_result role nty
                                                 (args_co `mkTransCo` first_co `mkTransCo` rest_co)
                                                 res_co) }
               _ -> -- No unique matching family instance exists;
                    -- we do not do anything
                    return (assemble_result role (mkTyConApp tc ntys) args_co res_co) }

      | otherwise
      = -- A synonym with no type families in the RHS; or data type etc
        -- Just normalise the arguments and rebuild
        do { (args_co, ntys, res_co) <- normalise_tc_args tc tys
           ; role <- getRole
           ; return (assemble_result role (mkTyConApp tc ntys) args_co res_co) }

      where
        assemble_result :: Role       -- r, ambient role in NormM monad
                        -> Type       -- nty, result type, possibly of changed kind
                        -> Coercion   -- orig_ty ~r nty, possibly heterogeneous
                        -> MCoercionN -- typeKind(orig_ty) ~N typeKind(nty)
                        -> (Coercion, Type)   -- (co :: orig_ty ~r nty_casted, nty_casted)
                                              -- where nty_casted has same kind as orig_ty
        assemble_result r nty orig_to_nty kind_co
          = ( final_co, nty_old_kind )
          where
            nty_old_kind = nty `mkCastTyMCo` mkSymMCo kind_co
            final_co     = mkCoherenceRightMCo r nty (mkSymMCo kind_co) orig_to_nty

    ---------------
    -- | Try to simplify a type-family application, by *one* step
    -- If topReduceTyFamApp_maybe env r F tys = Just (co, rhs, res_co)
    -- then    co     :: F tys ~R# rhs
    --         res_co :: typeKind(F tys) ~ typeKind(rhs)
    -- Type families and data families; always Representational role
    topReduceTyFamApp_maybe :: FamInstEnvs -> TyCon -> [Type]
                            -> Maybe (Coercion, Type, MCoercion)
    topReduceTyFamApp_maybe envs fam_tc arg_tys
      | isFamilyTyCon fam_tc   -- type families and data families
      , Just (co, rhs) <- reduceTyFamApp_maybe envs role fam_tc ntys
      = Just (args_co `mkTransCo` co, rhs, res_co)
      | otherwise
      = Nothing
      where
        role = Representational
        (args_co, ntys, res_co) = initNormM envs role (tyCoVarsOfTypes arg_tys) $
                                  normalise_tc_args fam_tc arg_tys

    normalise_tc_args :: TyCon -> [Type]             -- tc tys
                      -> NormM (Coercion, [Type], MCoercionN)
                      -- (co, new_tys), where
                      -- co :: tc tys ~ tc new_tys; might not be homogeneous
                      -- res_co :: typeKind(tc tys) ~N typeKind(tc new_tys)
    normalise_tc_args tc tys
      = do { role <- getRole
           ; (args_cos, nargs, res_co) <- normalise_args (tyConKind tc) (tyConRolesX role tc) tys
           ; return (mkTyConAppCo role tc args_cos, nargs, res_co) }

    ---------------
    normaliseType :: FamInstEnvs
                  -> Role  -- desired role of coercion
                  -> Type -> (Coercion, Type)
    normaliseType env role ty
      = initNormM env role (tyCoVarsOfType ty) $ normalise_type ty

    normalise_type :: Type                     -- old type
                   -> NormM (Coercion, Type)   -- (coercion, new type), where
                                               -- co :: old-type ~ new_type
    -- Normalise the input type, by eliminating *all* type-function redexes
    -- but *not* newtypes (which are visible to the programmer)
    -- Returns with Refl if nothing happens
    -- Does nothing to newtypes
    -- The returned coercion *must* be *homogeneous*
    -- See Note [Normalising types]
    -- Try not to disturb type synonyms if possible

    normalise_type ty
      = go ty
      where
        go (TyConApp tc tys) = normalise_tc_app tc tys
        go ty@(LitTy {})     = do { r <- getRole
                                  ; return (mkReflCo r ty, ty) }
        go (AppTy ty1 ty2) = go_app_tys ty1 [ty2]

        go ty@(FunTy { ft_mult = w, ft_arg = ty1, ft_res = ty2 })
          = do { (co1, nty1) <- go ty1
               ; (co2, nty2) <- go ty2
               ; (wco, wty) <- withRole Nominal $ go w
               ; r <- getRole
               ; return (mkFunCo r wco co1 co2, ty { ft_mult = wty, ft_arg = nty1, ft_res = nty2 }) }
        go (ForAllTy (Bndr tcvar vis) ty)
          = do { (lc', tv', h, ki') <- normalise_var_bndr tcvar
               ; (co, nty)          <- withLC lc' $ normalise_type ty
               ; let tv2 = setTyVarKind tv' ki'
               ; return (mkForAllCo tv' h co, ForAllTy (Bndr tv2 vis) nty) }
        go (TyVarTy tv)    = normalise_tyvar tv
        go (CastTy ty co)
          = do { (nco, nty) <- go ty
               ; lc <- getLC
               ; let co' = substRightCo lc co
               ; return (castCoercionKind2 nco Nominal ty nty co co'
                        , mkCastTy nty co') }
        go (CoercionTy co)
          = do { lc <- getLC
               ; r <- getRole
               ; let right_co = substRightCo lc co
               ; return ( mkProofIrrelCo r
                             (liftCoSubst Nominal lc (coercionType co))
                             co right_co
                        , mkCoercionTy right_co ) }

        go_app_tys :: Type   -- function
                   -> [Type] -- args
                   -> NormM (Coercion, Type)
        -- cf. GHC.Tc.Solver.Rewrite.rewrite_app_ty_args
        go_app_tys (AppTy ty1 ty2) tys = go_app_tys ty1 (ty2 : tys)
        go_app_tys fun_ty arg_tys
          = do { (fun_co, nfun) <- go fun_ty
               ; case tcSplitTyConApp_maybe nfun of
                   Just (tc, xis) ->
                     do { (second_co, nty) <- go (mkTyConApp tc (xis ++ arg_tys))
                       -- rewrite_app_ty_args avoids redundantly processing the xis,
                       -- but that's a much more performance-sensitive function.
                       -- This type normalisation is not called in a loop.
                        ; return (mkAppCos fun_co (map mkNomReflCo arg_tys) `mkTransCo` second_co, nty) }
                   Nothing ->
                     do { (args_cos, nargs, res_co) <- normalise_args (typeKind nfun)
                                                                      (repeat Nominal)
                                                                      arg_tys
                        ; role <- getRole
                        ; let nty = mkAppTys nfun nargs
                              nco = mkAppCos fun_co args_cos
                              nty_casted = nty `mkCastTyMCo` mkSymMCo res_co
                              final_co = mkCoherenceRightMCo role nty (mkSymMCo res_co) nco
                        ; return (final_co, nty_casted) } }

    normalise_args :: Kind    -- of the function
                   -> [Role]  -- roles at which to normalise args
                   -> [Type]  -- args
                   -> NormM ([Coercion], [Type], MCoercion)
    -- returns (cos, xis, res_co), where each xi is the normalised
    -- version of the corresponding type, each co is orig_arg ~ xi,
    -- and the res_co :: kind(f orig_args) ~ kind(f xis)
    -- NB: The xis might *not* have the same kinds as the input types,
    -- but the resulting application *will* be well-kinded
    -- cf. GHC.Tc.Solver.Rewrite.rewrite_args_slow
    normalise_args fun_ki roles args
      = do { normed_args <- zipWithM normalise1 roles args
           ; let (xis, cos, res_co) = simplifyArgsWorker ki_binders inner_ki fvs roles normed_args
           ; return (map mkSymCo cos, xis, mkSymMCo res_co) }
      where
        (ki_binders, inner_ki) = splitPiTys fun_ki
        fvs = tyCoVarsOfTypes args

        -- rewriter conventions are different from ours
        impedance_match :: NormM (Coercion, Type) -> NormM (Type, Coercion)
        impedance_match action = do { (co, ty) <- action
                                    ; return (ty, mkSymCo co) }

        normalise1 role ty
          = impedance_match $ withRole role $ normalise_type ty

    normalise_tyvar :: TyVar -> NormM (Coercion, Type)
    normalise_tyvar tv
      = ASSERT( isTyVar tv )
        do { lc <- getLC
           ; r  <- getRole
           ; return $ case liftCoSubstTyVar lc r tv of
               Just co -> (co, coercionRKind co)
               Nothing -> (mkReflCo r ty, ty) }
      where ty = mkTyVarTy tv

    normalise_var_bndr :: TyCoVar -> NormM (LiftingContext, TyCoVar, Coercion, Kind)
    normalise_var_bndr tcvar
      -- works for both tvar and covar
      = do { lc1 <- getLC
           ; env <- getEnv
           ; let callback lc ki = runNormM (normalise_type ki) env lc Nominal
           ; return $ liftCoSubstVarBndrUsing callback lc1 tcvar }

    -- | a monad for the normalisation functions, reading 'FamInstEnvs',
    -- a 'LiftingContext', and a 'Role'.
    newtype NormM a = NormM { runNormM ::
                                FamInstEnvs -> LiftingContext -> Role -> a }
        deriving (Functor)

    initNormM :: FamInstEnvs -> Role
              -> TyCoVarSet   -- the in-scope variables
              -> NormM a -> a
    initNormM env role vars (NormM thing_inside)
      = thing_inside env lc role
      where
        in_scope = mkInScopeSet vars
        lc       = emptyLiftingContext in_scope

    getRole :: NormM Role
    getRole = NormM (\ _ _ r -> r)

    getLC :: NormM LiftingContext
    getLC = NormM (\ _ lc _ -> lc)

    getEnv :: NormM FamInstEnvs
    getEnv = NormM (\ env _ _ -> env)

    withRole :: Role -> NormM a -> NormM a
    withRole r thing = NormM $ \ envs lc _old_r -> runNormM thing envs lc r

    withLC :: LiftingContext -> NormM a -> NormM a
    withLC lc thing = NormM $ \ envs _old_lc r -> runNormM thing envs lc r

    instance Monad NormM where
      ma >>= fmb = NormM $ \env lc r ->
                   let a = runNormM ma env lc r in
                   runNormM (fmb a) env lc r

    instance Applicative NormM where
      pure x = NormM $ \ _ _ _ -> x
      (<*>)  = ap