-- (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,

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

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

        lookupFamInstEnv, lookupFamInstEnvConflicts,

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

        -- Flattening
    ) 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)) ])
    ax = fi_axiom famInst

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

-- 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 }
     -- 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

*                                                                      *
*                                                                      *

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)

--  * 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
    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)
    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])
    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
   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)
    brs1 = coAxiomBranches ax1
    brs2 = coAxiomBranches ax2

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

*                                                                      *
*                                                                      *

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 }
    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 }
    (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 = [] }) }
    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

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

   = lookup_fam_inst_env match
     match _ tpl_tvs tpl_tys tys = tcMatchTys tpl_tvs tpl_tys tys

    :: 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
    (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 = []

    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

        (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
      = (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
      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
      = unwrapNewTypeStepper
        \ 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)

    (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)
    (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)

*                                                                      *
*                                                                      *

Note [Flattening]

As described in
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
    -- 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 []
    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
    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
    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