    -- (c) The University of Glasgow 2006
    --
    -- FamInstEnv: Type checked family instance declarations

    {-# LANGUAGE CPP, GADTs, ScopedTypeVariables #-}

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

            FamInstEnvs, FamInstEnv, emptyFamInstEnv, emptyFamInstEnvs,
            extendFamInstEnv, deleteFromFamInstEnv, extendFamInstEnvList,
            identicalFamInstHead, famInstEnvElts, familyInstances, orphNamesOfFamInst,

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

            FamInstMatch(..),
            lookupFamInstEnv, lookupFamInstEnvConflicts,
            isDominatedBy,

            -- Normalisation
            topNormaliseType, topNormaliseType_maybe,
            normaliseType, normaliseTcApp,
            reduceTyFamApp_maybe, chooseBranch,

            -- Flattening
            flattenTys
        ) where

    #include "HsVersions.h"

    import InstEnv
    import Unify
    import Type
    import TcType ( orphNamesOfTypes )
    import TypeRep
    import TyCon
    import Coercion
    import CoAxiom
    import VarSet
    import VarEnv
    import Name
    import UniqFM
    import Outputable
    import Maybes
    import TrieMap
    import Unique
    import Util
    import Var
    import Pair
    import SrcLoc
    import NameSet
    import FastString

    {-
    ************************************************************************
    *                                                                      *
              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
                , 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-match field] in InstEnv
                , fi_tcs   :: [Maybe Name]  -- Top of type args
                    -- INVARIANT: fi_tcs = roughMatchTcs fi_tys

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

                , fi_tys    :: [Type]       --   and its arg types
                    -- INVARIANT: fi_tvs = coAxiomTyVars fi_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

    -- 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

    -- Prints the FamInst as a family instance declaration
    -- NB: FamInstEnv.pprFamInst is used only for internal, debug printing
    --     See pprTyThing.pprFamInst for printing for the user
    pprFamInst :: FamInst -> SDoc
    pprFamInst famInst
      = hang (pprFamInstHdr famInst)
           2 (vcat [ ifPprDebug (ptext (sLit "Coercion axiom:") <+> ppr ax)
                   , ifPprDebug (ptext (sLit "RHS:") <+> ppr (famInstRHS famInst)) ])
      where
        ax = fi_axiom famInst

    pprFamInstHdr :: FamInst -> SDoc
    pprFamInstHdr fi@(FamInst {fi_flavor = flavor})
      = pprTyConSort <+> pp_instance <+> pp_head
      where
        -- For *associated* types, say "type T Int = blah"
        -- For *top level* type instances, say "type instance T Int = blah"
        pp_instance
          | isTyConAssoc fam_tc = empty
          | otherwise           = ptext (sLit "instance")

        (fam_tc, etad_lhs_tys) = famInstSplitLHS fi
        vanilla_pp_head = pprTypeApp fam_tc etad_lhs_tys

        pp_head | DataFamilyInst rep_tc <- flavor
                , isAlgTyCon rep_tc
                , let extra_tvs = dropList etad_lhs_tys (tyConTyVars rep_tc)
                , not (null extra_tvs)
                = getPprStyle $ \ sty ->
                  if debugStyle sty
                  then vanilla_pp_head   -- With -dppr-debug just show it as-is
                  else pprTypeApp fam_tc (etad_lhs_tys ++ mkTyVarTys extra_tvs)
                         -- Without -dppr-debug, eta-expand
                         -- See Trac #8674
                         -- (This is probably over the top now that we use this
                         --  only for internal debug printing; PprTyThing.pprFamInst
                         --  is used for user-level printing.)
                | otherwise
                = vanilla_pp_head

        pprTyConSort = case flavor of
                         SynFamilyInst        -> ptext (sLit "type")
                         DataFamilyInst tycon
                           | isDataTyCon     tycon -> ptext (sLit "data")
                           | isNewTyCon      tycon -> ptext (sLit "newtype")
                           | isAbstractTyCon tycon -> ptext (sLit "data")
                           | otherwise             -> ptext (sLit "WEIRD") <+> ppr tycon

    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 TcIface. 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
                      -> [Maybe Name]       -- 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_tys    = tys,
          fi_rhs    = rhs,
          fi_axiom  = axiom,
          fi_flavor = flavor }
      where
         -- See Note [Lazy axiom match]
         ~(CoAxiom { co_ax_branches =
           ~(FirstBranch ~(CoAxBranch { cab_lhs = tys
                                      , cab_tvs = tvs
                                      , cab_rhs = rhs })) }) = 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 contructors
    and pattern matching.

    Neverthless 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 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.  The reason
    for this eta-reduction is decribed in TcInstDcls
       Note [Eta reduction for data family axioms]
    -}

    type FamInstEnv = UniqFM FamilyInstEnv  -- Maps a family to its instances
         -- See Note [FamInstEnv]

    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) = ptext (sLit "FamIE") <+> vcat (map ppr fs)

    -- 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 = emptyUFM

    famInstEnvElts :: FamInstEnv -> [FamInst]
    famInstEnvElts fi = [elt | FamIE elts <- eltsUFM fi, elt <- elts]

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

    -- | Collects the names of the concrete types and type constructors that
    -- make up the LHS of a type family instance, including the family
    -- name itself.
    --
    -- For instance, given `type family Foo a b`:
    -- `type instance Foo (F (G (H a))) b = ...` would yield [Foo,F,G,H]
    --
    -- Used in the implementation of ":info" in GHCi.
    orphNamesOfFamInst :: FamInst -> NameSet
    orphNamesOfFamInst fam_inst
      = orphNamesOfTypes (concat (brListMap cab_lhs (coAxiomBranches axiom)))
        `extendNameSet` getName (coAxiomTyCon axiom)
      where
        axiom = fi_axiom fam_inst

    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})
      = addToUFM_C add inst_env cls_nm (FamIE [ins_item])
      where
        add (FamIE items) _ = FamIE (ins_item:items)

    deleteFromFamInstEnv :: FamInstEnv -> FamInst -> FamInstEnv
    -- Used only for overriding in GHCi
    deleteFromFamInstEnv inst_env fam_inst@(FamInst {fi_fam = fam_nm})
     = adjustUFM adjust inst_env fam_nm
     where
       adjust :: FamilyInstEnv -> FamilyInstEnv
       adjust (FamIE items)
         = FamIE (filterOut (identicalFamInstHead fam_inst) items)

    identicalFamInstHead :: FamInst -> FamInst -> Bool
    -- ^ True when the LHSs are identical
    -- Used for overriding in GHCi
    identicalFamInstHead (FamInst { fi_axiom = ax1 }) (FamInst { fi_axiom = ax2 })
      =  coAxiomTyCon ax1 == coAxiomTyCon ax2
      && brListLength brs1 == brListLength brs2
      && and (brListZipWith identical_branch brs1 brs2)
      where
        brs1 = coAxiomBranches ax1
        brs2 = coAxiomBranches ax2

        identical_branch br1 br2
          =  isJust (tcMatchTys tvs1 lhs1 lhs2)
          && isJust (tcMatchTys tvs2 lhs2 lhs1)
          where
            tvs1 = mkVarSet (coAxBranchTyVars br1)
            tvs2 = mkVarSet (coAxBranchTyVars br2)
            lhs1 = coAxBranchLHS br1
            lhs2 = coAxBranchLHS br2

    {-
    ************************************************************************
    *                                                                      *
                    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].)

    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 the
    potentially-overlapping group is closed.

    As another example, consider this:

    type family G x
    type instance 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).
    -}

    -- See Note [Compatibility]
    compatibleBranches :: CoAxBranch -> CoAxBranch -> Bool
    compatibleBranches (CoAxBranch { cab_lhs = lhs1, cab_rhs = rhs1 })
                       (CoAxBranch { cab_lhs = lhs2, cab_rhs = rhs2 })
      = case tcUnifyTysFG instanceBindFun lhs1 lhs2 of
          SurelyApart -> True
          Unifiable subst
            | Type.substTy subst rhs1 `eqType` Type.substTy subst rhs2
            -> True
          _ -> False

    -- takes a CoAxiom with unknown branch incompatibilities and computes
    -- the compatibilities
    -- See Note [Storing compatibility] in CoAxiom
    computeAxiomIncomps :: CoAxiom br -> CoAxiom br
    computeAxiomIncomps ax@(CoAxiom { co_ax_branches = branches })
      = ax { co_ax_branches = go [] branches }
      where
        go :: [CoAxBranch] -> BranchList CoAxBranch br -> BranchList CoAxBranch br
        go prev_branches (FirstBranch br)
          = FirstBranch (br { cab_incomps = mk_incomps br prev_branches })
        go prev_branches (NextBranch br tail)
          = let br' = br { cab_incomps = mk_incomps br prev_branches } in
            NextBranch br' (go (br' : prev_branches) tail)

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

    {-
    ************************************************************************
    *                                                                      *
               Constructing axioms
        These functions are here because tidyType / tcUnifyTysFG
        are not available in CoAxiom
    *                                                                      *
    ************************************************************************

    Note [Tidy axioms when we build them]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    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 Trac #7524.
    -}

    -- all axiom roles are Nominal, as this is only used with type families
    mkCoAxBranch :: [TyVar] -- original, possibly stale, tyvars
                 -> [Type]  -- LHS patterns
                 -> Type    -- RHS
                 -> SrcSpan
                 -> CoAxBranch
    mkCoAxBranch tvs lhs rhs loc
      = CoAxBranch { cab_tvs     = tvs1
                   , cab_lhs     = tidyTypes env lhs
                   , cab_roles   = map (const Nominal) tvs1
                   , cab_rhs     = tidyType  env rhs
                   , cab_loc     = loc
                   , cab_incomps = placeHolderIncomps }
      where
        (env, tvs1) = tidyTyVarBndrs emptyTidyEnv tvs
        -- See Note [Tidy axioms when we build them]

    -- 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
      = computeAxiomIncomps $
        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 = toBranchList 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 = FirstBranch (branch { cab_incomps = [] }) }

    mkSingleCoAxiom :: Name -> [TyVar] -> TyCon -> [Type] -> Type -> CoAxiom Unbranched
    mkSingleCoAxiom ax_name tvs 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     = Nominal
                , co_ax_implicit = False
                , co_ax_branches = FirstBranch (branch { cab_incomps = [] }) }
      where
        branch = mkCoAxBranch tvs lhs_tys rhs_ty (getSrcSpan ax_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]
                                     }
      -- See Note [Over-saturated matches]

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

    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_tvs tpl_tys tys = tcMatchTys tpl_tvs 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( tyVarsOfTypes 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

    {-
    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 TvSubst

    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) <- lookupUFM ie fam
      = find insts    -- The common case
      | otherwise = []
      where

        find [] = []
        find (item@(FamInst { fi_tcs = mb_tcs, fi_tvs = tpl_tvs,
                              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 })
            : 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 family axioms] in TcInstDcls.
    -}

    -- 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_tvs = tvs, cab_lhs = tys })
            = isJust $ tcMatchTys (mkVarSet tvs) 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* normlise the type arguments first, so this may not
    --     go as far as you want. If you want normalised type arguments,
    --     use normaliseTcArgs first.
    --
    -- The TyCon can be oversaturated.
    -- Works on both open and closed families

    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 = fam_inst
                     , fim_tys =      inst_tys } : _ <- lookupFamInstEnv envs tc tys
          -- NB: Allow multiple matches because of compatible overlap

      = let ax     = famInstAxiom fam_inst
            co     = mkUnbranchedAxInstCo role ax inst_tys
            ty     = pSnd (coercionKind co)
        in Just (co, ty)

      | Just ax <- isClosedSynFamilyTyCon_maybe tc
      , Just (ind, inst_tys) <- chooseBranch ax tys
      = let co     = mkAxInstCo role ax ind inst_tys
            ty     = pSnd (coercionKind co)
        in Just (co, ty)

      | Just ax           <- isBuiltInSynFamTyCon_maybe tc
      , Just (coax,ts,ty) <- sfMatchFam ax tys
      = let co = mkAxiomRuleCo coax ts []
        in Just (co, ty)

      | otherwise
      = Nothing

    -- The axiom can be oversaturated. (Closed families only.)
    chooseBranch :: CoAxiom Branched -> [Type] -> Maybe (BranchIndex, [Type])
    chooseBranch axiom tys
      = do { let num_pats = coAxiomNumPats axiom
                 (target_tys, extra_tys) = splitAt num_pats tys
                 branches = coAxiomBranches axiom
           ; (ind, inst_tys) <- findBranch (fromBranchList branches) 0 target_tys
           ; return (ind, inst_tys ++ extra_tys) }

    -- The axiom must *not* be oversaturated
    findBranch :: [CoAxBranch]             -- branches to check
               -> BranchIndex              -- index of current branch
               -> [Type]                   -- target types
               -> Maybe (BranchIndex, [Type])
    findBranch (CoAxBranch { cab_tvs = tpl_tvs, cab_lhs = tpl_lhs, cab_incomps = incomps }
                  : rest) ind target_tys
      = case tcMatchTys (mkVarSet tpl_tvs) tpl_lhs target_tys of
          Just subst -- matching worked. now, check for apartness.
            |  all (isSurelyApart
                    . tcUnifyTysFG instanceBindFun flattened_target
                    . coAxBranchLHS) incomps
            -> -- matching worked & we're apart from all incompatible branches. success
               Just (ind, substTyVars subst tpl_tvs)

          -- failure. keep looking
          _ -> findBranch rest (ind+1) target_tys

      where isSurelyApart SurelyApart = True
            isSurelyApart _           = False

            flattened_target = flattenTys in_scope target_tys
            in_scope = mkInScopeSet (unionVarSets $
                                     map (tyVarsOfTypes . coAxBranchLHS) incomps)

    -- fail if no branches left
    findBranch [] _ _ = Nothing

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

    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
    --      * newtypes
    -- using appropriate coercions.  Specifically, if
    --   topNormaliseType_maybe env ty = Maybe (co, ty')
    -- then
    --   (a) co :: ty ~ ty'
    --   (b) ty' is not a newtype, and is not a type-family redex
    --
    -- However, ty' can be something like (Maybe (F ty)), where
    -- (F ty) is a redex.
    --
    -- Its a bit like Type.repType, but handles type families too
    -- The coercion returned is always an R coercion

    topNormaliseType_maybe env ty
      = topNormaliseTypeX_maybe stepper ty
      where
        stepper
          = unwrapNewTypeStepper
            `composeSteppers`
            \ rec_nts tc tys ->
            let (args_co, ntys) = normaliseTcArgs env Representational tc tys in
            case reduceTyFamApp_maybe env Representational tc ntys of
              Just (co, rhs) -> NS_Step rec_nts rhs (args_co `mkTransCo` co)
              Nothing        -> NS_Done

    ---------------
    normaliseTcApp :: FamInstEnvs -> Role -> TyCon -> [Type] -> (Coercion, Type)
    -- See comments on normaliseType for the arguments of this function
    normaliseTcApp env role tc tys
      | isTypeSynonymTyCon tc
      , Just (tenv, rhs, ntys') <- tcExpandTyCon_maybe tc ntys
      , (co2, ninst_rhs) <- normaliseType env role (Type.substTy (mkTopTvSubst tenv) rhs)
      = if isReflCo co2 then (args_co,                 mkTyConApp tc ntys)
                        else (args_co `mkTransCo` co2, mkAppTys ninst_rhs ntys')

      | Just (first_co, ty') <- reduceTyFamApp_maybe env role tc ntys
      , (rest_co,nty) <- normaliseType env role ty'
      = (args_co `mkTransCo` first_co `mkTransCo` rest_co, nty)

      | otherwise   -- No unique matching family instance exists;
                    -- we do not do anything
      = (args_co, mkTyConApp tc ntys)

      where
        (args_co, ntys) = normaliseTcArgs env role tc tys

    ---------------
    normaliseTcArgs :: FamInstEnvs            -- environment with family instances
                     -> Role                   -- desired role of output coercion
                     -> TyCon -> [Type]        -- tc tys
                     -> (Coercion, [Type])     -- (co, new_tys), where
                                               -- co :: tc tys ~ tc new_tys
    normaliseTcArgs env role tc tys
      = (mkTyConAppCo role tc cois, ntys)
      where
        (cois, ntys) = zipWithAndUnzip (normaliseType env) (tyConRolesX role tc) tys

    ---------------
    normaliseType :: FamInstEnvs            -- environment with family instances
                   -> Role                   -- desired role of output coercion
                   -> Type                   -- old type
                   -> (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
    -- Try to not to disturb type syonyms if possible

    normaliseType env role (TyConApp tc tys)
      = normaliseTcApp env role tc tys
    normaliseType _env role ty@(LitTy {}) = (mkReflCo role ty, ty)
    normaliseType env role (AppTy ty1 ty2)
      = let (coi1,nty1) = normaliseType env role    ty1
            (coi2,nty2) = normaliseType env Nominal ty2
        in  (mkAppCo coi1 coi2, mkAppTy nty1 nty2)
    normaliseType env role (FunTy ty1 ty2)
      = let (coi1,nty1) = normaliseType env role ty1
            (coi2,nty2) = normaliseType env role ty2
        in  (mkFunCo role coi1 coi2, mkFunTy nty1 nty2)
    normaliseType env role (ForAllTy tyvar ty1)
      = let (coi,nty1) = normaliseType env role ty1
        in  (mkForAllCo tyvar coi, ForAllTy tyvar nty1)
    normaliseType _  role ty@(TyVarTy _)
      = (mkReflCo role ty,ty)

    {-
    ************************************************************************
    *                                                                      *
                  Flattening
    *                                                                      *
    ************************************************************************

    Note [Flattening]
    ~~~~~~~~~~~~~~~~~

    As described in
    http://research.microsoft.com/en-us/um/people/simonpj/papers/ext-f/axioms-extended.pdf
    we sometimes need to flatten core types before unifying them. Flattening
    means replacing all top-level uses of type functions with fresh variables,
    taking care to preserve sharing. That is, the type (Either (F a b) (F a b)) should
    flatten to (Either c c), never (Either c d).

    Defined here because of module dependencies.
    -}

    type FlattenMap = TypeMap TyVar

    -- See Note [Flattening]
    flattenTys :: InScopeSet -> [Type] -> [Type]
    flattenTys in_scope tys = snd $ coreFlattenTys all_in_scope emptyTypeMap tys
      where
        -- when we hit a type function, we replace it with a fresh variable
        -- but, we need to make sure that this fresh variable isn't mentioned
        -- *anywhere* in the types we're flattening, even if locally-bound in
        -- a forall. That way, we can ensure consistency both within and outside
        -- of that forall.
        all_in_scope = in_scope `extendInScopeSetSet` allTyVarsInTys tys

    coreFlattenTys :: InScopeSet -> FlattenMap -> [Type] -> (FlattenMap, [Type])
    coreFlattenTys in_scope = go []
      where
        go rtys m []         = (m, reverse rtys)
        go rtys m (ty : tys)
          = let (m', ty') = coreFlattenTy in_scope m ty in
            go (ty' : rtys) m' tys

    coreFlattenTy :: InScopeSet -> FlattenMap -> Type -> (FlattenMap, Type)
    coreFlattenTy in_scope = go
      where
        go m ty | Just ty' <- coreView ty = go m ty'

        go m ty@(TyVarTy {}) = (m, ty)
        go m (AppTy ty1 ty2) = let (m1, ty1') = go m  ty1
                                   (m2, ty2') = go m1 ty2 in
                               (m2, AppTy ty1' ty2')
        go m (TyConApp tc tys)
          | isFamilyTyCon tc
          = let (m', tv) = coreFlattenTyFamApp in_scope m tc tys in
            (m', mkTyVarTy tv)

          | otherwise
          = let (m', tys') = coreFlattenTys in_scope m tys in
            (m', mkTyConApp tc tys')

        go m (FunTy ty1 ty2) = let (m1, ty1') = go m  ty1
                                   (m2, ty2') = go m1 ty2 in
                               (m2, FunTy ty1' ty2')

          -- Note to RAE: this will have to be changed with kind families
        go m (ForAllTy tv ty) = let (m', ty') = go m ty in
                                (m', ForAllTy tv ty')

        go m ty@(LitTy {}) = (m, ty)

    coreFlattenTyFamApp :: InScopeSet -> FlattenMap
                        -> TyCon         -- type family tycon
                        -> [Type]        -- args
                        -> (FlattenMap, TyVar)
    coreFlattenTyFamApp in_scope m fam_tc fam_args
      = case lookupTypeMap m fam_ty of
          Just tv -> (m, tv)
                  -- we need fresh variables here, but this is called far from
                  -- any good source of uniques. So, we generate one from thin
                  -- air, using the arbitrary prime number 71 as a seed
          Nothing -> let tyvar_unique = deriveUnique (getUnique fam_tc) 71
                         tyvar_name   = mkSysTvName tyvar_unique (fsLit "fl")
                         tv = uniqAway in_scope $ mkTyVar tyvar_name (typeKind fam_ty)
                         m' = extendTypeMap m fam_ty tv in
                     (m', tv)
      where fam_ty = TyConApp fam_tc fam_args

    allTyVarsInTys :: [Type] -> VarSet
    allTyVarsInTys []       = emptyVarSet
    allTyVarsInTys (ty:tys) = allTyVarsInTy ty `unionVarSet` allTyVarsInTys tys

    allTyVarsInTy :: Type -> VarSet
    allTyVarsInTy = go
      where
        go (TyVarTy tv)      = unitVarSet tv
        go (AppTy ty1 ty2)   = (go ty1) `unionVarSet` (go ty2)
        go (TyConApp _ tys)  = allTyVarsInTys tys
        go (FunTy ty1 ty2)   = (go ty1) `unionVarSet` (go ty2)
        go (ForAllTy tv ty)  = (go (tyVarKind tv)) `unionVarSet`
                               unitVarSet tv `unionVarSet`
                               (go ty) -- don't remove tv
        go (LitTy {})        = emptyVarSet