types/OptCoercion.hs

    -- (c) The University of Glasgow 2006

    {-# LANGUAGE CPP #-}

    module OptCoercion ( optCoercion, checkAxInstCo ) where

    #include "HsVersions.h"

    import Coercion
    import Type hiding( substTyVarBndr, substTy, extendTvSubst )
    import TcType       ( exactTyVarsOfType )
    import TyCon
    import CoAxiom
    import Var
    import VarSet
    import FamInstEnv   ( flattenTys )
    import VarEnv
    import StaticFlags      ( opt_NoOptCoercion )
    import Outputable
    import Pair
    import FastString
    import Util
    import Unify
    import ListSetOps
    import InstEnv
    import Control.Monad   ( zipWithM )

    {-
    ************************************************************************
    *                                                                      *
                     Optimising coercions
    *                                                                      *
    ************************************************************************

    Note [Subtle shadowing in coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Supose we optimising a coercion
        optCoercion (forall (co_X5:t1~t2). ...co_B1...)
    The co_X5 is a wild-card; the bound variable of a coercion for-all
    should never appear in the body of the forall. Indeed we often
    write it like this
        optCoercion ( (t1~t2) => ...co_B1... )

    Just because it's a wild-card doesn't mean we are free to choose
    whatever variable we like.  For example it'd be wrong for optCoercion
    to return
       forall (co_B1:t1~t2). ...co_B1...
    because now the co_B1 (which is really free) has been captured, and
    subsequent substitutions will go wrong.  That's why we can't use
    mkCoPredTy in the ForAll case, where this note appears.

    Note [Optimising coercion optimisation]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Looking up a coercion's role or kind is linear in the size of the
    coercion. Thus, doing this repeatedly during the recursive descent
    of coercion optimisation is disastrous. We must be careful to avoid
    doing this if at all possible.

    Because it is generally easy to know a coercion's components' roles
    from the role of the outer coercion, we pass down the known role of
    the input in the algorithm below. We also keep functions opt_co2
    and opt_co3 separate from opt_co4, so that the former two do Phantom
    checks that opt_co4 can avoid. This is a big win because Phantom coercions
    rarely appear within non-phantom coercions -- only in some TyConAppCos
    and some AxiomInstCos. We handle these cases specially by calling
    opt_co2.
    -}

    optCoercion :: CvSubst -> Coercion -> NormalCo
    -- ^ optCoercion applies a substitution to a coercion,
    --   *and* optimises it to reduce its size
    optCoercion env co
      | opt_NoOptCoercion = substCo env co
      | otherwise         = opt_co1 env False co

    type NormalCo = Coercion
      -- Invariants:
      --  * The substitution has been fully applied
      --  * For trans coercions (co1 `trans` co2)
      --       co1 is not a trans, and neither co1 nor co2 is identity
      --  * If the coercion is the identity, it has no CoVars of CoTyCons in it (just types)

    type NormalNonIdCo = NormalCo  -- Extra invariant: not the identity

    -- | Do we apply a @sym@ to the result?
    type SymFlag = Bool

    -- | Do we force the result to be representational?
    type ReprFlag = Bool

    -- | Optimize a coercion, making no assumptions.
    opt_co1 :: CvSubst
            -> SymFlag
            -> Coercion -> NormalCo
    opt_co1 env sym co = opt_co2 env sym (coercionRole co) co
    {-
    opt_co env sym co
     = pprTrace "opt_co {" (ppr sym <+> ppr co $$ ppr env) $
       co1 `seq`
       pprTrace "opt_co done }" (ppr co1) $
       (WARN( not same_co_kind, ppr co  <+> dcolon <+> ppr (coercionType co)
                             $$ ppr co1 <+> dcolon <+> ppr (coercionType co1) )
        WARN( not (coreEqCoercion co1 simple_result),
               (text "env=" <+> ppr env) $$
               (text "input=" <+> ppr co) $$
               (text "simple=" <+> ppr simple_result) $$
               (text "opt=" <+> ppr co1) )
       co1)
     where
       co1 = opt_co' env sym co
       same_co_kind = s1 `eqType` s2 && t1 `eqType` t2
       Pair s t = coercionKind (substCo env co)
       (s1,t1) | sym = (t,s)
               | otherwise = (s,t)
       Pair s2 t2 = coercionKind co1

       simple_result | sym = mkSymCo (substCo env co)
                     | otherwise = substCo env co
    -}

    -- See Note [Optimising coercion optimisation]
    -- | Optimize a coercion, knowing the coercion's role. No other assumptions.
    opt_co2 :: CvSubst
            -> SymFlag
            -> Role   -- ^ The role of the input coercion
            -> Coercion -> NormalCo
    opt_co2 env sym Phantom co = opt_phantom env sym co
    opt_co2 env sym r       co = opt_co3 env sym Nothing r co

    -- See Note [Optimising coercion optimisation]
    -- | Optimize a coercion, knowing the coercion's non-Phantom role.
    opt_co3 :: CvSubst -> SymFlag -> Maybe Role -> Role -> Coercion -> NormalCo
    opt_co3 env sym (Just Phantom) _ co = opt_phantom env sym co
    opt_co3 env sym (Just Representational) r co = opt_co4 env sym True  r co
      -- if mrole is Just Nominal, that can't be a downgrade, so we can ignore
    opt_co3 env sym _                       r co = opt_co4 env sym False r co

    -- See Note [Optimising coercion optimisation]
    -- | Optimize a non-phantom coercion.
    opt_co4 :: CvSubst -> SymFlag -> ReprFlag -> Role -> Coercion -> NormalCo

    opt_co4 env _   rep r (Refl _r ty)
      = ASSERT( r == _r )
        Refl (chooseRole rep r) (substTy env ty)

    opt_co4 env sym rep r (SymCo co)  = opt_co4 env (not sym) rep r co

    opt_co4 env sym rep r g@(TyConAppCo _r tc cos)
      = ASSERT( r == _r )
        case (rep, r) of
          (True, Nominal) ->
            mkTyConAppCo Representational tc
                         (zipWith3 (opt_co3 env sym)
                                   (map Just (tyConRolesX Representational tc))
                                   (repeat Nominal)
                                   cos)
          (False, Nominal) ->
            mkTyConAppCo Nominal tc (map (opt_co4 env sym False Nominal) cos)
          (_, Representational) ->
                          -- must use opt_co2 here, because some roles may be P
                          -- See Note [Optimising coercion optimisation]
            mkTyConAppCo r tc (zipWith (opt_co2 env sym)
                                       (tyConRolesX r tc)  -- the current roles
                                       cos)
          (_, Phantom) -> pprPanic "opt_co4 sees a phantom!" (ppr g)

    opt_co4 env sym rep r (AppCo co1 co2) = mkAppCo (opt_co4 env sym rep r co1)
                                                    (opt_co4 env sym False Nominal co2)
    opt_co4 env sym rep r (ForAllCo tv co)
      = case substTyVarBndr env tv of
          (env', tv') -> mkForAllCo tv' (opt_co4 env' sym rep r co)
         -- Use the "mk" functions to check for nested Refls

    opt_co4 env sym rep r (CoVarCo cv)
      | Just co <- lookupCoVar env cv
      = opt_co4 (zapCvSubstEnv env) sym rep r co

      | Just cv1 <- lookupInScope (getCvInScope env) cv
      = ASSERT( isCoVar cv1 ) wrapRole rep r $ wrapSym sym (CoVarCo cv1)
                    -- cv1 might have a substituted kind!

      | otherwise = WARN( True, ptext (sLit "opt_co: not in scope:") <+> ppr cv $$ ppr env)
                    ASSERT( isCoVar cv )
                    wrapRole rep r $ wrapSym sym (CoVarCo cv)

    opt_co4 env sym rep r (AxiomInstCo con ind cos)
        -- Do *not* push sym inside top-level axioms
        -- e.g. if g is a top-level axiom
        --   g a : f a ~ a
        -- then (sym (g ty)) /= g (sym ty) !!
      = ASSERT( r == coAxiomRole con )
        wrapRole rep (coAxiomRole con) $
        wrapSym sym $
                           -- some sub-cos might be P: use opt_co2
                           -- See Note [Optimising coercion optimisation]
        AxiomInstCo con ind (zipWith (opt_co2 env False)
                                     (coAxBranchRoles (coAxiomNthBranch con ind))
                                     cos)
          -- Note that the_co does *not* have sym pushed into it

    opt_co4 env sym rep r (UnivCo s _r oty1 oty2)
      = ASSERT( r == _r )
        opt_univ env s (chooseRole rep r) a b
      where
        (a,b) = if sym then (oty2,oty1) else (oty1,oty2)

    opt_co4 env sym rep r (TransCo co1 co2)
                          -- sym (g `o` h) = sym h `o` sym g
      | sym       = opt_trans in_scope co2' co1'
      | otherwise = opt_trans in_scope co1' co2'
      where
        co1' = opt_co4 env sym rep r co1
        co2' = opt_co4 env sym rep r co2
        in_scope = getCvInScope env

    opt_co4 env sym rep r co@(NthCo {}) = opt_nth_co env sym rep r co

    opt_co4 env sym rep r (LRCo lr co)
      | Just pr_co <- splitAppCo_maybe co
      = ASSERT( r == Nominal )
        opt_co4 env sym rep Nominal (pickLR lr pr_co)
      | Just pr_co <- splitAppCo_maybe co'
      = ASSERT( r == Nominal )
        if rep
        then opt_co4 (zapCvSubstEnv env) False True Nominal (pickLR lr pr_co)
        else pickLR lr pr_co
      | otherwise
      = wrapRole rep Nominal $ LRCo lr co'
      where
        co' = opt_co4 env sym False Nominal co

    opt_co4 env sym rep r (InstCo co ty)
        -- See if the first arg is already a forall
        -- ...then we can just extend the current substitution
      | Just (tv, co_body) <- splitForAllCo_maybe co
      = opt_co4 (extendTvSubst env tv ty') sym rep r co_body

         -- See if it is a forall after optimization
         -- If so, do an inefficient one-variable substitution
      | Just (tv, co'_body) <- splitForAllCo_maybe co'
      = substCoWithTy (getCvInScope env) tv ty' co'_body

      | otherwise = InstCo co' ty'
      where
        co' = opt_co4 env sym rep r co
        ty' = substTy env ty

    opt_co4 env sym _ r (SubCo co)
      = ASSERT( r == Representational )
        opt_co4 env sym True Nominal co

    -- XXX: We could add another field to CoAxiomRule that
    -- would allow us to do custom simplifications.
    opt_co4 env sym rep r (AxiomRuleCo co ts cs)
      = ASSERT( r == coaxrRole co )
        wrapRole rep r $
        wrapSym sym $
        AxiomRuleCo co (map (substTy env) ts)
                       (zipWith (opt_co2 env False) (coaxrAsmpRoles co) cs)

    -------------
    -- | Optimize a phantom coercion. The input coercion may not necessarily
    -- be a phantom, but the output sure will be.
    opt_phantom :: CvSubst -> SymFlag -> Coercion -> NormalCo
    opt_phantom env sym co
      = if sym
        then opt_univ env (fsLit "opt_phantom") Phantom ty2 ty1
        else opt_univ env (fsLit "opt_phantom") Phantom ty1 ty2
      where
        Pair ty1 ty2 = coercionKind co

    opt_univ :: CvSubst -> FastString -> Role -> Type -> Type -> Coercion
    opt_univ env prov role oty1 oty2
      | Just (tc1, tys1) <- splitTyConApp_maybe oty1
      , Just (tc2, tys2) <- splitTyConApp_maybe oty2
      , tc1 == tc2
      = mkTyConAppCo role tc1 (zipWith3 (opt_univ env prov) (tyConRolesX role tc1) tys1 tys2)

      | Just (l1, r1) <- splitAppTy_maybe oty1
      , Just (l2, r2) <- splitAppTy_maybe oty2
      , typeKind l1 `eqType` typeKind l2   -- kind(r1) == kind(r2) by consequence
      = let role' = if role == Phantom then Phantom else Nominal in
           -- role' is to comform to mkAppCo's precondition
        mkAppCo (opt_univ env prov role l1 l2) (opt_univ env prov role' r1 r2)

      | Just (tv1, ty1) <- splitForAllTy_maybe oty1
      , Just (tv2, ty2) <- splitForAllTy_maybe oty2
      , tyVarKind tv1 `eqType` tyVarKind tv2  -- rule out a weird unsafeCo
      = case substTyVarBndr2 env tv1 tv2 of { (env1, env2, tv') ->
        let ty1' = substTy env1 ty1
            ty2' = substTy env2 ty2 in
        mkForAllCo tv' (opt_univ (zapCvSubstEnv2 env1 env2) prov role ty1' ty2') }

      | otherwise
      = mkUnivCo prov role (substTy env oty1) (substTy env oty2)

    -------------
    -- NthCo must be handled separately, because it's the one case where we can't
    -- tell quickly what the component coercion's role is from the containing
    -- coercion. To avoid repeated coercionRole calls as opt_co1 calls opt_co2,
    -- we just look for nested NthCo's, which can happen in practice.
    opt_nth_co :: CvSubst -> SymFlag -> ReprFlag -> Role -> Coercion -> NormalCo
    opt_nth_co env sym rep r = go []
      where
        go ns (NthCo n co) = go (n:ns) co
          -- previous versions checked if the tycon is decomposable. This
          -- is redundant, because a non-decomposable tycon under an NthCo
          -- is entirely bogus. See docs/core-spec/core-spec.pdf.
        go ns co
          = opt_nths ns co

          -- input coercion is *not* yet sym'd or opt'd
        opt_nths [] co = opt_co4 env sym rep r co
        opt_nths (n:ns) (TyConAppCo _ _ cos) = opt_nths ns (cos `getNth` n)

          -- here, the co isn't a TyConAppCo, so we opt it, hoping to get
          -- a TyConAppCo as output. We don't know the role, so we use
          -- opt_co1. This is slightly annoying, because opt_co1 will call
          -- coercionRole, but as long as we don't have a long chain of
          -- NthCo's interspersed with some other coercion former, we should
          -- be OK.
        opt_nths ns co = opt_nths' ns (opt_co1 env sym co)

          -- input coercion *is* sym'd and opt'd
        opt_nths' [] co
          = if rep && (r == Nominal)
                -- propagate the SubCo:
            then opt_co4 (zapCvSubstEnv env) False True r co
            else co
        opt_nths' (n:ns) (TyConAppCo _ _ cos) = opt_nths' ns (cos `getNth` n)
        opt_nths' ns co = wrapRole rep r (mk_nths ns co)

        mk_nths [] co = co
        mk_nths (n:ns) co = mk_nths ns (mkNthCo n co)

    -------------
    opt_transList :: InScopeSet -> [NormalCo] -> [NormalCo] -> [NormalCo]
    opt_transList is = zipWith (opt_trans is)

    opt_trans :: InScopeSet -> NormalCo -> NormalCo -> NormalCo
    opt_trans is co1 co2
      | isReflCo co1 = co2
      | otherwise    = opt_trans1 is co1 co2

    opt_trans1 :: InScopeSet -> NormalNonIdCo -> NormalCo -> NormalCo
    -- First arg is not the identity
    opt_trans1 is co1 co2
      | isReflCo co2 = co1
      | otherwise    = opt_trans2 is co1 co2

    opt_trans2 :: InScopeSet -> NormalNonIdCo -> NormalNonIdCo -> NormalCo
    -- Neither arg is the identity
    opt_trans2 is (TransCo co1a co1b) co2
        -- Don't know whether the sub-coercions are the identity
      = opt_trans is co1a (opt_trans is co1b co2)

    opt_trans2 is co1 co2
      | Just co <- opt_trans_rule is co1 co2
      = co

    opt_trans2 is co1 (TransCo co2a co2b)
      | Just co1_2a <- opt_trans_rule is co1 co2a
      = if isReflCo co1_2a
        then co2b
        else opt_trans1 is co1_2a co2b

    opt_trans2 _ co1 co2
      = mkTransCo co1 co2

    ------
    -- Optimize coercions with a top-level use of transitivity.
    opt_trans_rule :: InScopeSet -> NormalNonIdCo -> NormalNonIdCo -> Maybe NormalCo

    -- Push transitivity through matching destructors
    opt_trans_rule is in_co1@(NthCo d1 co1) in_co2@(NthCo d2 co2)
      | d1 == d2
      , co1 `compatible_co` co2
      = fireTransRule "PushNth" in_co1 in_co2 $
        mkNthCo d1 (opt_trans is co1 co2)

    opt_trans_rule is in_co1@(LRCo d1 co1) in_co2@(LRCo d2 co2)
      | d1 == d2
      , co1 `compatible_co` co2
      = fireTransRule "PushLR" in_co1 in_co2 $
        mkLRCo d1 (opt_trans is co1 co2)

    -- Push transitivity inside instantiation
    opt_trans_rule is in_co1@(InstCo co1 ty1) in_co2@(InstCo co2 ty2)
      | ty1 `eqType` ty2
      , co1 `compatible_co` co2
      = fireTransRule "TrPushInst" in_co1 in_co2 $
        mkInstCo (opt_trans is co1 co2) ty1

    -- Push transitivity down through matching top-level constructors.
    opt_trans_rule is in_co1@(TyConAppCo r1 tc1 cos1) in_co2@(TyConAppCo r2 tc2 cos2)
      | tc1 == tc2
      = ASSERT( r1 == r2 )
        fireTransRule "PushTyConApp" in_co1 in_co2 $
        TyConAppCo r1 tc1 (opt_transList is cos1 cos2)

    opt_trans_rule is in_co1@(AppCo co1a co1b) in_co2@(AppCo co2a co2b)
      = fireTransRule "TrPushApp" in_co1 in_co2 $
        mkAppCo (opt_trans is co1a co2a) (opt_trans is co1b co2b)

    -- Eta rules
    opt_trans_rule is co1@(TyConAppCo r tc cos1) co2
      | Just cos2 <- etaTyConAppCo_maybe tc co2
      = ASSERT( length cos1 == length cos2 )
        fireTransRule "EtaCompL" co1 co2 $
        TyConAppCo r tc (opt_transList is cos1 cos2)

    opt_trans_rule is co1 co2@(TyConAppCo r tc cos2)
      | Just cos1 <- etaTyConAppCo_maybe tc co1
      = ASSERT( length cos1 == length cos2 )
        fireTransRule "EtaCompR" co1 co2 $
        TyConAppCo r tc (opt_transList is cos1 cos2)

    opt_trans_rule is co1@(AppCo co1a co1b) co2
      | Just (co2a,co2b) <- etaAppCo_maybe co2
      = fireTransRule "EtaAppL" co1 co2 $
        mkAppCo (opt_trans is co1a co2a) (opt_trans is co1b co2b)

    opt_trans_rule is co1 co2@(AppCo co2a co2b)
      | Just (co1a,co1b) <- etaAppCo_maybe co1
      = fireTransRule "EtaAppR" co1 co2 $
        mkAppCo (opt_trans is co1a co2a) (opt_trans is co1b co2b)

    -- Push transitivity inside forall
    opt_trans_rule is co1 co2
      | Just (tv1,r1) <- splitForAllCo_maybe co1
      , Just (tv2,r2) <- etaForAllCo_maybe co2
      , let r2' = substCoWithTy is' tv2 (mkTyVarTy tv1) r2
            is' = is `extendInScopeSet` tv1
      = fireTransRule "EtaAllL" co1 co2 $
        mkForAllCo tv1 (opt_trans2 is' r1 r2')

      | Just (tv2,r2) <- splitForAllCo_maybe co2
      , Just (tv1,r1) <- etaForAllCo_maybe co1
      , let r1' = substCoWithTy is' tv1 (mkTyVarTy tv2) r1
            is' = is `extendInScopeSet` tv2
      = fireTransRule "EtaAllR" co1 co2 $
        mkForAllCo tv1 (opt_trans2 is' r1' r2)

    -- Push transitivity inside axioms
    opt_trans_rule is co1 co2

      -- See Note [Why call checkAxInstCo during optimisation]
      -- TrPushSymAxR
      | Just (sym, con, ind, cos1) <- co1_is_axiom_maybe
      , Just cos2 <- matchAxiom sym con ind co2
      , True <- sym
      , let newAxInst = AxiomInstCo con ind (opt_transList is (map mkSymCo cos2) cos1)
      , Nothing <- checkAxInstCo newAxInst
      = fireTransRule "TrPushSymAxR" co1 co2 $ SymCo newAxInst

      -- TrPushAxR
      | Just (sym, con, ind, cos1) <- co1_is_axiom_maybe
      , Just cos2 <- matchAxiom sym con ind co2
      , False <- sym
      , let newAxInst = AxiomInstCo con ind (opt_transList is cos1 cos2)
      , Nothing <- checkAxInstCo newAxInst
      = fireTransRule "TrPushAxR" co1 co2 newAxInst

      -- TrPushSymAxL
      | Just (sym, con, ind, cos2) <- co2_is_axiom_maybe
      , Just cos1 <- matchAxiom (not sym) con ind co1
      , True <- sym
      , let newAxInst = AxiomInstCo con ind (opt_transList is cos2 (map mkSymCo cos1))
      , Nothing <- checkAxInstCo newAxInst
      = fireTransRule "TrPushSymAxL" co1 co2 $ SymCo newAxInst

      -- TrPushAxL
      | Just (sym, con, ind, cos2) <- co2_is_axiom_maybe
      , Just cos1 <- matchAxiom (not sym) con ind co1
      , False <- sym
      , let newAxInst = AxiomInstCo con ind (opt_transList is cos1 cos2)
      , Nothing <- checkAxInstCo newAxInst
      = fireTransRule "TrPushAxL" co1 co2 newAxInst

      -- TrPushAxSym/TrPushSymAx
      | Just (sym1, con1, ind1, cos1) <- co1_is_axiom_maybe
      , Just (sym2, con2, ind2, cos2) <- co2_is_axiom_maybe
      , con1 == con2
      , ind1 == ind2
      , sym1 == not sym2
      , let branch = coAxiomNthBranch con1 ind1
            qtvs = coAxBranchTyVars branch
            lhs  = coAxNthLHS con1 ind1
            rhs  = coAxBranchRHS branch
            pivot_tvs = exactTyVarsOfType (if sym2 then rhs else lhs)
      , all (`elemVarSet` pivot_tvs) qtvs
      = fireTransRule "TrPushAxSym" co1 co2 $
        if sym2
        then liftCoSubstWith role qtvs (opt_transList is cos1 (map mkSymCo cos2)) lhs  -- TrPushAxSym
        else liftCoSubstWith role qtvs (opt_transList is (map mkSymCo cos1) cos2) rhs  -- TrPushSymAx
      where
        co1_is_axiom_maybe = isAxiom_maybe co1
        co2_is_axiom_maybe = isAxiom_maybe co2
        role = coercionRole co1 -- should be the same as coercionRole co2!

    opt_trans_rule _ co1 co2        -- Identity rule
      | (Pair ty1 _, r) <- coercionKindRole co1
      , Pair _ ty2 <- coercionKind co2
      , ty1 `eqType` ty2
      = fireTransRule "RedTypeDirRefl" co1 co2 $
        Refl r ty2

    opt_trans_rule _ _ _ = Nothing

    fireTransRule :: String -> Coercion -> Coercion -> Coercion -> Maybe Coercion
    fireTransRule _rule _co1 _co2 res
      = -- pprTrace ("Trans rule fired: " ++ _rule) (vcat [ppr _co1, ppr _co2, ppr res]) $
        Just res

    {-
    Note [Conflict checking with AxiomInstCo]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider the following type family and axiom:

    type family Equal (a :: k) (b :: k) :: Bool
    type instance where
      Equal a a = True
      Equal a b = False
    --
    Equal :: forall k::BOX. k -> k -> Bool
    axEqual :: { forall k::BOX. forall a::k. Equal k a a ~ True
               ; forall k::BOX. forall a::k. forall b::k. Equal k a b ~ False }

    We wish to disallow (axEqual[1] <*> <Int> <Int). (Recall that the index is
    0-based, so this is the second branch of the axiom.) The problem is that, on
    the surface, it seems that (axEqual[1] <*> <Int> <Int>) :: (Equal * Int Int ~
    False) and that all is OK. But, all is not OK: we want to use the first branch
    of the axiom in this case, not the second. The problem is that the parameters
    of the first branch can unify with the supplied coercions, thus meaning that
    the first branch should be taken. See also Note [Branched instance checking]
    in types/FamInstEnv.lhs.

    Note [Why call checkAxInstCo during optimisation]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    It is possible that otherwise-good-looking optimisations meet with disaster
    in the presence of axioms with multiple equations. Consider

    type family Equal (a :: *) (b :: *) :: Bool where
      Equal a a = True
      Equal a b = False
    type family Id (a :: *) :: * where
      Id a = a

    axEq :: { [a::*].       Equal a a ~ True
            ; [a::*, b::*]. Equal a b ~ False }
    axId :: [a::*]. Id a ~ a

    co1 = Equal (axId[0] Int) (axId[0] Bool)
      :: Equal (Id Int) (Id Bool) ~  Equal Int Bool
    co2 = axEq[1] <Int> <Bool>
      :: Equal Int Bool ~ False

    We wish to optimise (co1 ; co2). We end up in rule TrPushAxL, noting that
    co2 is an axiom and that matchAxiom succeeds when looking at co1. But, what
    happens when we push the coercions inside? We get

    co3 = axEq[1] (axId[0] Int) (axId[0] Bool)
      :: Equal (Id Int) (Id Bool) ~ False

    which is bogus! This is because the type system isn't smart enough to know
    that (Id Int) and (Id Bool) are Surely Apart, as they're headed by type
    families. At the time of writing, I (Richard Eisenberg) couldn't think of
    a way of detecting this any more efficient than just building the optimised
    coercion and checking.
    -}

    -- | Check to make sure that an AxInstCo is internally consistent.
    -- Returns the conflicting branch, if it exists
    -- See Note [Conflict checking with AxiomInstCo]
    checkAxInstCo :: Coercion -> Maybe CoAxBranch
    -- defined here to avoid dependencies in Coercion
    -- If you edit this function, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in CoreLint
    checkAxInstCo (AxiomInstCo ax ind cos)
      = let branch   = coAxiomNthBranch ax ind
            tvs      = coAxBranchTyVars branch
            incomps  = coAxBranchIncomps branch
            tys      = map (pFst . coercionKind) cos
            subst    = zipOpenTvSubst tvs tys
            target   = Type.substTys subst (coAxBranchLHS branch)
            in_scope = mkInScopeSet $
                       unionVarSets (map (tyVarsOfTypes . coAxBranchLHS) incomps)
            flattened_target = flattenTys in_scope target in
        check_no_conflict flattened_target incomps
      where
        check_no_conflict :: [Type] -> [CoAxBranch] -> Maybe CoAxBranch
        check_no_conflict _    [] = Nothing
        check_no_conflict flat (b@CoAxBranch { cab_lhs = lhs_incomp } : rest)
             -- See Note [Apartness] in FamInstEnv
          | SurelyApart <- tcUnifyTysFG instanceBindFun flat lhs_incomp
          = check_no_conflict flat rest
          | otherwise
          = Just b
    checkAxInstCo _ = Nothing

    -----------
    wrapSym :: SymFlag -> Coercion -> Coercion
    wrapSym sym co | sym       = SymCo co
                   | otherwise = co

    -- | Conditionally set a role to be representational
    wrapRole :: ReprFlag
             -> Role         -- ^ current role
             -> Coercion -> Coercion
    wrapRole False _       = id
    wrapRole True  current = downgradeRole Representational current

    -- | If we require a representational role, return that. Otherwise,
    -- return the "default" role provided.
    chooseRole :: ReprFlag
               -> Role    -- ^ "default" role
               -> Role
    chooseRole True _ = Representational
    chooseRole _    r = r
    -----------
    -- takes two tyvars and builds env'ts to map them to the same tyvar
    substTyVarBndr2 :: CvSubst -> TyVar -> TyVar
                    -> (CvSubst, CvSubst, TyVar)
    substTyVarBndr2 env tv1 tv2
      = case substTyVarBndr env tv1 of
          (env1, tv1') -> (env1, extendTvSubstAndInScope env tv2 (mkTyVarTy tv1'), tv1')

    zapCvSubstEnv2 :: CvSubst -> CvSubst -> CvSubst
    zapCvSubstEnv2 env1 env2 = mkCvSubst (is1 `unionInScope` is2) []
      where is1 = getCvInScope env1
            is2 = getCvInScope env2
    -----------
    isAxiom_maybe :: Coercion -> Maybe (Bool, CoAxiom Branched, Int, [Coercion])
    isAxiom_maybe (SymCo co)
      | Just (sym, con, ind, cos) <- isAxiom_maybe co
      = Just (not sym, con, ind, cos)
    isAxiom_maybe (AxiomInstCo con ind cos)
      = Just (False, con, ind, cos)
    isAxiom_maybe _ = Nothing

    matchAxiom :: Bool -- True = match LHS, False = match RHS
               -> CoAxiom br -> Int -> Coercion -> Maybe [Coercion]
    -- If we succeed in matching, then *all the quantified type variables are bound*
    -- E.g.   if tvs = [a,b], lhs/rhs = [b], we'll fail
    matchAxiom sym ax@(CoAxiom { co_ax_tc = tc }) ind co
      = let (CoAxBranch { cab_tvs   = qtvs
                        , cab_roles = roles
                        , cab_lhs   = lhs
                        , cab_rhs   = rhs }) = coAxiomNthBranch ax ind in
        case liftCoMatch (mkVarSet qtvs) (if sym then (mkTyConApp tc lhs) else rhs) co of
          Nothing    -> Nothing
          Just subst -> zipWithM (liftCoSubstTyVar subst) roles qtvs

    -------------
    compatible_co :: Coercion -> Coercion -> Bool
    -- Check whether (co1 . co2) will be well-kinded
    compatible_co co1 co2
      = x1 `eqType` x2
      where
        Pair _ x1 = coercionKind co1
        Pair x2 _ = coercionKind co2

    -------------
    etaForAllCo_maybe :: Coercion -> Maybe (TyVar, Coercion)
    -- Try to make the coercion be of form (forall tv. co)
    etaForAllCo_maybe co
      | Just (tv, r) <- splitForAllCo_maybe co
      = Just (tv, r)

      | Pair ty1 ty2  <- coercionKind co
      , Just (tv1, _) <- splitForAllTy_maybe ty1
      , Just (tv2, _) <- splitForAllTy_maybe ty2
      , tyVarKind tv1 `eqKind` tyVarKind tv2
      = Just (tv1, mkInstCo co (mkTyVarTy tv1))

      | otherwise
      = Nothing

    etaAppCo_maybe :: Coercion -> Maybe (Coercion,Coercion)
    -- If possible, split a coercion
    --   g :: t1a t1b ~ t2a t2b
    -- into a pair of coercions (left g, right g)
    etaAppCo_maybe co
      | Just (co1,co2) <- splitAppCo_maybe co
      = Just (co1,co2)
      | (Pair ty1 ty2, Nominal) <- coercionKindRole co
      , Just (_,t1) <- splitAppTy_maybe ty1
      , Just (_,t2) <- splitAppTy_maybe ty2
      , typeKind t1 `eqType` typeKind t2      -- Note [Eta for AppCo]
      = Just (LRCo CLeft co, LRCo CRight co)
      | otherwise
      = Nothing

    etaTyConAppCo_maybe :: TyCon -> Coercion -> Maybe [Coercion]
    -- If possible, split a coercion
    --       g :: T s1 .. sn ~ T t1 .. tn
    -- into [ Nth 0 g :: s1~t1, ..., Nth (n-1) g :: sn~tn ]
    etaTyConAppCo_maybe tc (TyConAppCo _ tc2 cos2)
      = ASSERT( tc == tc2 ) Just cos2

    etaTyConAppCo_maybe tc co
      | isDecomposableTyCon tc
      , Pair ty1 ty2     <- coercionKind co
      , Just (tc1, tys1) <- splitTyConApp_maybe ty1
      , Just (tc2, tys2) <- splitTyConApp_maybe ty2
      , tc1 == tc2
      , let n = length tys1
      = ASSERT( tc == tc1 )
        ASSERT( n == length tys2 )
        Just (decomposeCo n co)
        -- NB: n might be <> tyConArity tc
        -- e.g.   data family T a :: * -> *
        --        g :: T a b ~ T c d

      | otherwise
      = Nothing

    {-
    Note [Eta for AppCo]
    ~~~~~~~~~~~~~~~~~~~~
    Suppose we have
       g :: s1 t1 ~ s2 t2

    Then we can't necessarily make
       left  g :: s1 ~ s2
       right g :: t1 ~ t2
    because it's possible that
       s1 :: * -> *         t1 :: *
       s2 :: (*->*) -> *    t2 :: * -> *
    and in that case (left g) does not have the same
    kind on either side.

    It's enough to check that
      kind t1 = kind t2
    because if g is well-kinded then
      kind (s1 t2) = kind (s2 t2)
    and these two imply
      kind s1 = kind s2
    -}