-- (c) The University of Glasgow 2006

{-# LANGUAGE ScopedTypeVariables, PatternSynonyms, MultiWayIf #-}

{-# LANGUAGE DeriveFunctor #-}

module GHC.Core.Unify (
        tcMatchTy, tcMatchTyKi,
        tcMatchTys, tcMatchTyKis,
        tcMatchTyX, tcMatchTysX, tcMatchTyKisX,
        tcMatchTyX_BM, ruleMatchTyKiX,

        -- Side-effect free unification
        tcUnifyTy, tcUnifyTyKi, tcUnifyTys, tcUnifyTyKis,
        tcUnifyTysFG, tcUnifyTyWithTFs,
        BindFun, BindFlag(..), matchBindFun, alwaysBindFun,
        UnifyResult, UnifyResultM(..), MaybeApartReason(..),
        typesCantMatch, typesAreApart,

        -- Matching a type against a lifted type (coercion)
        liftCoMatch,

        -- The core flattening algorithm
        flattenTys, flattenTysX,

   ) where

import GHC.Prelude

import GHC.Types.Var
import GHC.Types.Var.Env
import GHC.Types.Var.Set
import GHC.Types.Name( Name, mkSysTvName, mkSystemVarName )
import GHC.Builtin.Names( tYPETyConKey, cONSTRAINTTyConKey )
import GHC.Core.Type     hiding ( getTvSubstEnv )
import GHC.Core.Coercion hiding ( getCvSubstEnv )
import GHC.Core.TyCon
import GHC.Core.TyCo.Rep
import GHC.Core.TyCo.Compare ( eqType, tcEqType )
import GHC.Core.TyCo.FVs     ( tyCoVarsOfCoList, tyCoFVsOfTypes )
import GHC.Core.TyCo.Subst   ( mkTvSubst, emptyIdSubstEnv )
import GHC.Core.Map.Type
import GHC.Utils.FV( FV, fvVarList )
import GHC.Utils.Misc
import GHC.Data.Pair
import GHC.Utils.Outputable
import GHC.Types.Unique
import GHC.Types.Unique.FM
import GHC.Types.Unique.Set
import GHC.Exts( oneShot )
import GHC.Utils.Panic
import GHC.Utils.Panic.Plain
import GHC.Data.FastString

import Data.List ( mapAccumL )
import Control.Monad
import qualified Data.Semigroup as S

{-

Unification is much tricker than you might think.

1. The substitution we generate binds the *template type variables*
   which are given to us explicitly.

2. We want to match in the presence of foralls;
        e.g     (forall a. t1) ~ (forall b. t2)

   That is what the RnEnv2 is for; it does the alpha-renaming
   that makes it as if a and b were the same variable.
   Initialising the RnEnv2, so that it can generate a fresh
   binder when necessary, entails knowing the free variables of
   both types.

3. We must be careful not to bind a template type variable to a
   locally bound variable.  E.g.
        (forall a. x) ~ (forall b. b)
   where x is the template type variable.  Then we do not want to
   bind x to a/b!  This is a kind of occurs check.
   The necessary locals accumulate in the RnEnv2.

Note [tcMatchTy vs tcMatchTyKi]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This module offers two variants of matching: with kinds and without.
The TyKi variant takes two types, of potentially different kinds,
and matches them. Along the way, it necessarily also matches their
kinds. The Ty variant instead assumes that the kinds are already
eqType and so skips matching up the kinds.

How do you choose between them?

1. If you know that the kinds of the two types are eqType, use
   the Ty variant. It is more efficient, as it does less work.

2. If the kinds of variables in the template type might mention type families,
   use the Ty variant (and do other work to make sure the kinds
   work out). These pure unification functions do a straightforward
   syntactic unification and do no complex reasoning about type
   families. Note that the types of the variables in instances can indeed
   mention type families, so instance lookup must use the Ty variant.

   (Nothing goes terribly wrong -- no panics -- if there might be type
   families in kinds in the TyKi variant. You just might get match
   failure even though a reducing a type family would lead to success.)

3. Otherwise, if you're sure that the variable kinds do not mention
   type families and you're not already sure that the kind of the template
   equals the kind of the target, then use the TyKi version.
-}

-- | Some unification functions are parameterised by a 'BindFun', which
-- says whether or not to allow a certain unification to take place.
-- A 'BindFun' takes the 'TyVar' involved along with the 'Type' it will
-- potentially be bound to.
--
-- It is possible for the variable to actually be a coercion variable
-- (Note [Matching coercion variables]), but only when one-way matching.
-- In this case, the 'Type' will be a 'CoercionTy'.
type BindFun = TyCoVar -> Type -> BindFlag

-- | @tcMatchTy t1 t2@ produces a substitution (over fvs(t1))
-- @s@ such that @s(t1)@ equals @t2@.
-- The returned substitution might bind coercion variables,
-- if the variable is an argument to a GADT constructor.
--
-- Precondition: typeKind ty1 `eqType` typeKind ty2
--
-- We don't pass in a set of "template variables" to be bound
-- by the match, because tcMatchTy (and similar functions) are
-- always used on top-level types, so we can bind any of the
-- free variables of the LHS.
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTy :: Type -> Type -> Maybe Subst
tcMatchTy ty1 ty2 = tcMatchTys [ty1] [ty2]

tcMatchTyX_BM :: BindFun -> Subst
              -> Type -> Type -> Maybe Subst
tcMatchTyX_BM bind_me subst ty1 ty2
  = tc_match_tys_x bind_me False subst [ty1] [ty2]

-- | Like 'tcMatchTy', but allows the kinds of the types to differ,
-- and thus matches them as well.
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTyKi :: Type -> Type -> Maybe Subst
tcMatchTyKi ty1 ty2
  = tc_match_tys alwaysBindFun True [ty1] [ty2]

-- | This is similar to 'tcMatchTy', but extends a substitution
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTyX :: Subst            -- ^ Substitution to extend
           -> Type                -- ^ Template
           -> Type                -- ^ Target
           -> Maybe Subst
tcMatchTyX subst ty1 ty2
  = tc_match_tys_x alwaysBindFun False subst [ty1] [ty2]

-- | Like 'tcMatchTy' but over a list of types.
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTys :: [Type]         -- ^ Template
           -> [Type]         -- ^ Target
           -> Maybe Subst    -- ^ One-shot; in principle the template
                             -- variables could be free in the target
tcMatchTys tys1 tys2
  = tc_match_tys alwaysBindFun False tys1 tys2

-- | Like 'tcMatchTyKi' but over a list of types.
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTyKis :: [Type]         -- ^ Template
             -> [Type]         -- ^ Target
             -> Maybe Subst -- ^ One-shot substitution
tcMatchTyKis tys1 tys2
  = tc_match_tys alwaysBindFun True tys1 tys2

-- | Like 'tcMatchTys', but extending a substitution
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTysX :: Subst       -- ^ Substitution to extend
            -> [Type]         -- ^ Template
            -> [Type]         -- ^ Target
            -> Maybe Subst -- ^ One-shot substitution
tcMatchTysX subst tys1 tys2
  = tc_match_tys_x alwaysBindFun False subst tys1 tys2

-- | Like 'tcMatchTyKis', but extending a substitution
-- See also Note [tcMatchTy vs tcMatchTyKi]
tcMatchTyKisX :: Subst        -- ^ Substitution to extend
              -> [Type]          -- ^ Template
              -> [Type]          -- ^ Target
              -> Maybe Subst  -- ^ One-shot substitution
tcMatchTyKisX subst tys1 tys2
  = tc_match_tys_x alwaysBindFun True subst tys1 tys2

-- | Same as tc_match_tys_x, but starts with an empty substitution
tc_match_tys :: BindFun
             -> Bool          -- ^ match kinds?
             -> [Type]
             -> [Type]
             -> Maybe Subst
tc_match_tys bind_me match_kis tys1 tys2
  = tc_match_tys_x bind_me match_kis (mkEmptySubst in_scope) tys1 tys2
  where
    in_scope = mkInScopeSet (tyCoVarsOfTypes tys1 `unionVarSet` tyCoVarsOfTypes tys2)

-- | Worker for 'tcMatchTysX' and 'tcMatchTyKisX'
tc_match_tys_x :: BindFun
               -> Bool          -- ^ match kinds?
               -> Subst
               -> [Type]
               -> [Type]
               -> Maybe Subst
tc_match_tys_x bind_me match_kis (Subst in_scope id_env tv_env cv_env) tys1 tys2
  = case tc_unify_tys bind_me
                      False  -- Matching, not unifying
                      False  -- Not an injectivity check
                      match_kis
                      (mkRnEnv2 in_scope) tv_env cv_env tys1 tys2 of
      Unifiable (tv_env', cv_env')
        -> Just $ Subst in_scope id_env tv_env' cv_env'
      _ -> Nothing

-- | This one is called from the expression matcher,
-- which already has a MatchEnv in hand
ruleMatchTyKiX
  :: TyCoVarSet          -- ^ template variables
  -> RnEnv2
  -> TvSubstEnv          -- ^ type substitution to extend
  -> Type                -- ^ Template
  -> Type                -- ^ Target
  -> Maybe TvSubstEnv
ruleMatchTyKiX tmpl_tvs rn_env tenv tmpl target
-- See Note [Kind coercions in Unify]
  = case tc_unify_tys (matchBindFun tmpl_tvs) False False
                      True -- <-- this means to match the kinds
                      rn_env tenv emptyCvSubstEnv [tmpl] [target] of
      Unifiable (tenv', _) -> Just tenv'
      _                    -> Nothing

-- | Allow binding only for any variable in the set. Variables may
-- be bound to any type.
-- Used when doing simple matching; e.g. can we find a substitution
--
-- @
-- S = [a :-> t1, b :-> t2] such that
--     S( Maybe (a, b->Int )  =   Maybe (Bool, Char -> Int)
-- @
matchBindFun :: TyCoVarSet -> BindFun
matchBindFun tvs tv _ty
  | tv `elemVarSet` tvs = BindMe
  | otherwise           = Apart

-- | Allow the binding of any variable to any type
alwaysBindFun :: BindFun
alwaysBindFun _tv _ty = BindMe

{-
************************************************************************
*                                                                      *
                GADTs
*                                                                      *
************************************************************************

Note [Pruning dead case alternatives]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider        data T a where
                   T1 :: T Int
                   T2 :: T a

                newtype X = MkX Int
                newtype Y = MkY Char

                type family F a
                type instance F Bool = Int

Now consider    case x of { T1 -> e1; T2 -> e2 }

The question before the house is this: if I know something about the type
of x, can I prune away the T1 alternative?

Suppose x::T Char.  It's impossible to construct a (T Char) using T1,
        Answer = YES we can prune the T1 branch (clearly)

Suppose x::T (F a), where 'a' is in scope.  Then 'a' might be instantiated
to 'Bool', in which case x::T Int, so
        ANSWER = NO (clearly)

We see here that we want precisely the apartness check implemented within
tcUnifyTysFG. So that's what we do! Two types cannot match if they are surely
apart. Note that since we are simply dropping dead code, a conservative test
suffices.
-}

-- | Given a list of pairs of types, are any two members of a pair surely
-- apart, even after arbitrary type function evaluation and substitution?
typesCantMatch :: [(Type,Type)] -> Bool
-- See Note [Pruning dead case alternatives]
typesCantMatch prs = any (uncurry typesAreApart) prs

typesAreApart :: Type -> Type -> Bool
typesAreApart t1 t2 = case tcUnifyTysFG alwaysBindFun [t1] [t2] of
                        SurelyApart -> True
                        _           -> False
{-
************************************************************************
*                                                                      *
             Unification
*                                                                      *
************************************************************************

Note [Fine-grained unification]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Do the types (x, x) and ([y], y) unify? The answer is seemingly "no" --
no substitution to finite types makes these match. But, a substitution to
*infinite* types can unify these two types: [x |-> [[[...]]], y |-> [[[...]]] ].
Why do we care? Consider these two type family instances:

type instance F x x   = Int
type instance F [y] y = Bool

If we also have

type instance Looper = [Looper]

then the instances potentially overlap. The solution is to use unification
over infinite terms. This is possible (see [1] for lots of gory details), but
a full algorithm is a little more power than we need. Instead, we make a
conservative approximation and just omit the occurs check.

[1]: http://research.microsoft.com/en-us/um/people/simonpj/papers/ext-f/axioms-extended.pdf

tcUnifyTys considers an occurs-check problem as the same as general unification
failure.

tcUnifyTysFG ("fine-grained") returns one of three results: success, occurs-check
failure ("MaybeApart"), or general failure ("SurelyApart").

See also #8162.

It's worth noting that unification in the presence of infinite types is not
complete. This means that, sometimes, a closed type family does not reduce
when it should. See test case indexed-types/should_fail/Overlap15 for an
example.

Note [Unification result]
~~~~~~~~~~~~~~~~~~~~~~~~~
When unifying t1 ~ t2, we return
* Unifiable s, if s is a substitution such that s(t1) is syntactically the
  same as s(t2), modulo type-synonym expansion.
* SurelyApart, if there is no substitution s such that s(t1) = s(t2),
  where "=" includes type-family reductions.
* MaybeApart mar s, when we aren't sure. `mar` is a MaybeApartReason.

Examples
* [a] ~ Maybe b: SurelyApart, because [] and Maybe can't unify
* [(a,Int)] ~ [(Bool,b)]:  Unifiable
* [F Int] ~ [Bool]: MaybeApart MARTypeFamily, because F Int might reduce to Bool (the unifier
                    does not try this)
* a ~ Maybe a: MaybeApart MARInfinite. Not Unifiable clearly, but not SurelyApart either; consider
       a := Loop
       where  type family Loop where Loop = Maybe Loop

There is the possibility that two types are MaybeApart for *both* reasons:

* (a, F Int) ~ (Maybe a, Bool)

What reason should we use? The *only* consumer of the reason is described
in Note [Infinitary substitution in lookup] in GHC.Core.InstEnv. The goal
there is identify which instances might match a target later (but don't
match now) -- except that we want to ignore the possibility of infinitary
substitutions. So let's examine a concrete scenario:

  class C a b c
  instance C a (Maybe a) Bool
  -- other instances, including one that will actually match
  [W] C b b (F Int)

Do we want the instance as a future possibility? No. The only way that
instance can match is in the presence of an infinite type (infinitely
nested Maybes). We thus say that MARInfinite takes precedence, so that
InstEnv treats this case as an infinitary substitution case; the fact
that a type family is involved is only incidental. We thus define
the Semigroup instance for MaybeApartReason to prefer MARInfinite.

Note [The substitution in MaybeApart]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The constructor MaybeApart carries data with it, typically a TvSubstEnv. Why?
Because consider unifying these:

(a, a, Int) ~ (b, [b], Bool)

If we go left-to-right, we start with [a |-> b]. Then, on the middle terms, we
apply the subst we have so far and discover that we need [b |-> [b]]. Because
this fails the occurs check, we say that the types are MaybeApart (see above
Note [Fine-grained unification]). But, we can't stop there! Because if we
continue, we discover that Int is SurelyApart from Bool, and therefore the
types are apart. This has practical consequences for the ability for closed
type family applications to reduce. See test case
indexed-types/should_compile/Overlap14.

-}

-- | Simple unification of two types; all type variables are bindable
-- Precondition: the kinds are already equal
tcUnifyTy :: Type -> Type       -- All tyvars are bindable
          -> Maybe Subst
                       -- A regular one-shot (idempotent) substitution
tcUnifyTy t1 t2 = tcUnifyTys alwaysBindFun [t1] [t2]

-- | Like 'tcUnifyTy', but also unifies the kinds
tcUnifyTyKi :: Type -> Type -> Maybe Subst
tcUnifyTyKi t1 t2 = tcUnifyTyKis alwaysBindFun [t1] [t2]

-- | Unify two types, treating type family applications as possibly unifying
-- with anything and looking through injective type family applications.
-- Precondition: kinds are the same
tcUnifyTyWithTFs :: Bool  -- ^ True <=> do two-way unification;
                          --   False <=> do one-way matching.
                          --   See end of sec 5.2 from the paper
                 -> InScopeSet     -- Should include the free tyvars of both Type args
                 -> Type -> Type   -- Types to unify
                 -> Maybe Subst
-- This algorithm is an implementation of the "Algorithm U" presented in
-- the paper "Injective type families for Haskell", Figures 2 and 3.
-- The code is incorporated with the standard unifier for convenience, but
-- its operation should match the specification in the paper.
tcUnifyTyWithTFs twoWay in_scope t1 t2
  = case tc_unify_tys alwaysBindFun twoWay True False
                       rn_env emptyTvSubstEnv emptyCvSubstEnv
                       [t1] [t2] of
      Unifiable          (tv_subst, _cv_subst) -> Just $ maybe_fix tv_subst
      MaybeApart _reason (tv_subst, _cv_subst) -> Just $ maybe_fix tv_subst
      -- we want to *succeed* in questionable cases. This is a
      -- pre-unification algorithm.
      SurelyApart      -> Nothing
  where
    rn_env   = mkRnEnv2 in_scope

    maybe_fix | twoWay    = niFixSubst in_scope
              | otherwise = mkTvSubst in_scope -- when matching, don't confuse
                                               -- domain with range

-----------------
tcUnifyTys :: BindFun
           -> [Type] -> [Type]
           -> Maybe Subst
                                -- ^ A regular one-shot (idempotent) substitution
                                -- that unifies the erased types. See comments
                                -- for 'tcUnifyTysFG'

-- The two types may have common type variables, and indeed do so in the
-- second call to tcUnifyTys in GHC.Tc.Instance.FunDeps.checkClsFD
tcUnifyTys bind_fn tys1 tys2
  = case tcUnifyTysFG bind_fn tys1 tys2 of
      Unifiable result -> Just result
      _                -> Nothing

-- | Like 'tcUnifyTys' but also unifies the kinds
tcUnifyTyKis :: BindFun
             -> [Type] -> [Type]
             -> Maybe Subst
tcUnifyTyKis bind_fn tys1 tys2
  = case tcUnifyTyKisFG bind_fn tys1 tys2 of
      Unifiable result -> Just result
      _                -> Nothing

-- This type does double-duty. It is used in the UM (unifier monad) and to
-- return the final result. See Note [Fine-grained unification]
type UnifyResult = UnifyResultM Subst

-- | See Note [Unification result]
data UnifyResultM a = Unifiable a        -- the subst that unifies the types
                    | MaybeApart MaybeApartReason
                                 a       -- the subst has as much as we know
                                         -- it must be part of a most general unifier
                                         -- See Note [The substitution in MaybeApart]
                    | SurelyApart
                    deriving Functor

-- | Why are two types 'MaybeApart'? 'MARInfinite' takes precedence:
-- This is used (only) in Note [Infinitary substitution in lookup] in GHC.Core.InstEnv
-- As of Feb 2022, we never differentiate between MARTypeFamily and MARTypeVsConstraint;
-- it's really only MARInfinite that's interesting here.
data MaybeApartReason
  = MARTypeFamily   -- ^ matching e.g. F Int ~? Bool

  | MARInfinite     -- ^ matching e.g. a ~? Maybe a

  | MARTypeVsConstraint  -- ^ matching Type ~? Constraint or the arrow types
    -- See Note [Type and Constraint are not apart] in GHC.Builtin.Types.Prim

instance Outputable MaybeApartReason where
  ppr MARTypeFamily       = text "MARTypeFamily"
  ppr MARInfinite         = text "MARInfinite"
  ppr MARTypeVsConstraint = text "MARTypeVsConstraint"

instance Semigroup MaybeApartReason where
  -- see end of Note [Unification result] for why
  MARTypeFamily       <> r = r
  MARInfinite         <> _ = MARInfinite
  MARTypeVsConstraint <> r = r

instance Applicative UnifyResultM where
  pure  = Unifiable
  (<*>) = ap

instance Monad UnifyResultM where
  SurelyApart  >>= _ = SurelyApart
  MaybeApart r1 x >>= f = case f x of
                            Unifiable y     -> MaybeApart r1 y
                            MaybeApart r2 y -> MaybeApart (r1 S.<> r2) y
                            SurelyApart     -> SurelyApart
  Unifiable x  >>= f = f x

-- | @tcUnifyTysFG bind_tv tys1 tys2@ attempts to find a substitution @s@ (whose
-- domain elements all respond 'BindMe' to @bind_tv@) such that
-- @s(tys1)@ and that of @s(tys2)@ are equal, as witnessed by the returned
-- Coercions. This version requires that the kinds of the types are the same,
-- if you unify left-to-right.
tcUnifyTysFG :: BindFun
             -> [Type] -> [Type]
             -> UnifyResult
tcUnifyTysFG bind_fn tys1 tys2
  = tc_unify_tys_fg False bind_fn tys1 tys2

tcUnifyTyKisFG :: BindFun
               -> [Type] -> [Type]
               -> UnifyResult
tcUnifyTyKisFG bind_fn tys1 tys2
  = tc_unify_tys_fg True bind_fn tys1 tys2

tc_unify_tys_fg :: Bool
                -> BindFun
                -> [Type] -> [Type]
                -> UnifyResult
tc_unify_tys_fg match_kis bind_fn tys1 tys2
  = do { (env, _) <- tc_unify_tys bind_fn True False match_kis rn_env
                                  emptyTvSubstEnv emptyCvSubstEnv
                                  tys1 tys2
       ; return $ niFixSubst in_scope env }
  where
    in_scope = mkInScopeSet $ tyCoVarsOfTypes tys1 `unionVarSet` tyCoVarsOfTypes tys2
    rn_env   = mkRnEnv2 in_scope

-- | This function is actually the one to call the unifier -- a little
-- too general for outside clients, though.
tc_unify_tys :: BindFun
             -> AmIUnifying -- ^ True <=> unify; False <=> match
             -> Bool        -- ^ True <=> doing an injectivity check
             -> Bool        -- ^ True <=> treat the kinds as well
             -> RnEnv2
             -> TvSubstEnv  -- ^ substitution to extend
             -> CvSubstEnv
             -> [Type] -> [Type]
             -> UnifyResultM (TvSubstEnv, CvSubstEnv)
-- NB: It's tempting to ASSERT here that, if we're not matching kinds, then
-- the kinds of the types should be the same. However, this doesn't work,
-- as the types may be a dependent telescope, where later types have kinds
-- that mention variables occurring earlier in the list of types. Here's an
-- example (from typecheck/should_fail/T12709):
--   template: [rep :: RuntimeRep,       a :: TYPE rep]
--   target:   [LiftedRep :: RuntimeRep, Int :: TYPE LiftedRep]
-- We can see that matching the first pair will make the kinds of the second
-- pair equal. Yet, we still don't need a separate pass to unify the kinds
-- of these types, so it's appropriate to use the Ty variant of unification.
-- See also Note [tcMatchTy vs tcMatchTyKi].
tc_unify_tys bind_fn unif inj_check match_kis rn_env tv_env cv_env tys1 tys2
  = initUM tv_env cv_env $
    do { when match_kis $
         unify_tys env kis1 kis2
       ; unify_tys env tys1 tys2
       ; (,) <$> getTvSubstEnv <*> getCvSubstEnv }
  where
    env = UMEnv { um_bind_fun = bind_fn
                , um_skols    = emptyVarSet
                , um_unif     = unif
                , um_inj_tf   = inj_check
                , um_rn_env   = rn_env }

    kis1 = map typeKind tys1
    kis2 = map typeKind tys2

instance Outputable a => Outputable (UnifyResultM a) where
  ppr SurelyApart      = text "SurelyApart"
  ppr (Unifiable x)    = text "Unifiable" <+> ppr x
  ppr (MaybeApart r x) = text "MaybeApart" <+> ppr r <+> ppr x

{-
************************************************************************
*                                                                      *
                Non-idempotent substitution
*                                                                      *
************************************************************************

Note [Non-idempotent substitution]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
During unification we use a TvSubstEnv/CvSubstEnv pair that is
  (a) non-idempotent
  (b) loop-free; ie repeatedly applying it yields a fixed point

Note [Finding the substitution fixpoint]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Finding the fixpoint of a non-idempotent substitution arising from a
unification is much trickier than it looks, because of kinds.  Consider
   T k (H k (f:k)) ~ T * (g:*)
If we unify, we get the substitution
   [ k -> *
   , g -> H k (f:k) ]
To make it idempotent we don't want to get just
   [ k -> *
   , g -> H * (f:k) ]
We also want to substitute inside f's kind, to get
   [ k -> *
   , g -> H k (f:*) ]
If we don't do this, we may apply the substitution to something,
and get an ill-formed type, i.e. one where typeKind will fail.
This happened, for example, in #9106.

It gets worse.  In #14164 we wanted to take the fixpoint of
this substitution
   [ xs_asV :-> F a_aY6 (z_aY7 :: a_aY6)
                        (rest_aWF :: G a_aY6 (z_aY7 :: a_aY6))
   , a_aY6  :-> a_aXQ ]

We have to apply the substitution for a_aY6 two levels deep inside
the invocation of F!  We don't have a function that recursively
applies substitutions inside the kinds of variable occurrences (and
probably rightly so).

So, we work as follows:

 1. Start with the current substitution (which we are
    trying to fixpoint
       [ xs :-> F a (z :: a) (rest :: G a (z :: a))
       , a  :-> b ]

 2. Take all the free vars of the range of the substitution:
       {a, z, rest, b}
    NB: the free variable finder closes over
    the kinds of variable occurrences

 3. If none are in the domain of the substitution, stop.
    We have found a fixpoint.

 4. Remove the variables that are bound by the substitution, leaving
       {z, rest, b}

 5. Do a topo-sort to put them in dependency order:
       [ b :: *, z :: a, rest :: G a z ]

 6. Apply the substitution left-to-right to the kinds of these
    tyvars, extending it each time with a new binding, so we
    finish up with
       [ xs   :-> ..as before..
       , a    :-> b
       , b    :-> b    :: *
       , z    :-> z    :: b
       , rest :-> rest :: G b (z :: b) ]
    Note that rest now has the right kind

 7. Apply this extended substitution (once) to the range of
    the /original/ substitution.  (Note that we do the
    extended substitution would go on forever if you tried
    to find its fixpoint, because it maps z to z.)

 8. And go back to step 1

In Step 6 we use the free vars from Step 2 as the initial
in-scope set, because all of those variables appear in the
range of the substitution, so they must all be in the in-scope
set.  But NB that the type substitution engine does not look up
variables in the in-scope set; it is used only to ensure no
shadowing.
-}

niFixSubst :: InScopeSet -> TvSubstEnv -> Subst
-- Find the idempotent fixed point of the non-idempotent substitution
-- This is surprisingly tricky:
--   see Note [Finding the substitution fixpoint]
-- ToDo: use laziness instead of iteration?
niFixSubst in_scope tenv
  | not_fixpoint = niFixSubst in_scope (mapVarEnv (substTy subst) tenv)
  | otherwise    = subst
  where
    range_fvs :: FV
    range_fvs = tyCoFVsOfTypes (nonDetEltsUFM tenv)
          -- It's OK to use nonDetEltsUFM here because the
          -- order of range_fvs, range_tvs is immaterial

    range_tvs :: [TyVar]
    range_tvs = fvVarList range_fvs

    not_fixpoint  = any in_domain range_tvs
    in_domain tv  = tv `elemVarEnv` tenv

    free_tvs = scopedSort (filterOut in_domain range_tvs)

    -- See Note [Finding the substitution fixpoint], Step 6
    subst = foldl' add_free_tv
                  (mkTvSubst in_scope tenv)
                  free_tvs

    add_free_tv :: Subst -> TyVar -> Subst
    add_free_tv subst tv
      = extendTvSubst subst tv (mkTyVarTy tv')
     where
        tv' = updateTyVarKind (substTy subst) tv

niSubstTvSet :: TvSubstEnv -> TyCoVarSet -> TyCoVarSet
-- Apply the non-idempotent substitution to a set of type variables,
-- remembering that the substitution isn't necessarily idempotent
-- This is used in the occurs check, before extending the substitution
niSubstTvSet tsubst tvs
  = nonDetStrictFoldUniqSet (unionVarSet . get) emptyVarSet tvs
  -- It's OK to use a non-deterministic fold here because we immediately forget
  -- the ordering by creating a set.
  where
    get tv
      | Just ty <- lookupVarEnv tsubst tv
      = niSubstTvSet tsubst (tyCoVarsOfType ty)

      | otherwise
      = unitVarSet tv

{-
************************************************************************
*                                                                      *
                unify_ty: the main workhorse
*                                                                      *
************************************************************************

Note [Specification of unification]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The pure unifier, unify_ty, defined in this module, tries to work out
a substitution to make two types say True to eqType. NB: eqType is
itself not purely syntactic; it accounts for CastTys;
see Note [Non-trivial definitional equality] in GHC.Core.TyCo.Rep

Unlike the "impure unifiers" in the typechecker (the eager unifier in
GHC.Tc.Utils.Unify, and the constraint solver itself in GHC.Tc.Solver.Equality),
the pure unifier does /not/ work up to ~.

The algorithm implemented here is rather delicate, and we depend on it
to uphold certain properties. This is a summary of these required
properties.

Notation:
 θ,φ  substitutions
 ξ    type-function-free types
 τ,σ  other types
 τ♭   type τ, flattened

 ≡    eqType

(U1) Soundness.
     If (unify τ₁ τ₂) = Unifiable θ, then θ(τ₁) ≡ θ(τ₂).
     θ is a most general unifier for τ₁ and τ₂.

(U2) Completeness.
     If (unify ξ₁ ξ₂) = SurelyApart,
     then there exists no substitution θ such that θ(ξ₁) ≡ θ(ξ₂).

These two properties are stated as Property 11 in the "Closed Type Families"
paper (POPL'14). Below, this paper is called [CTF].

(U3) Apartness under substitution.
     If (unify ξ τ♭) = SurelyApart, then (unify ξ θ(τ)♭) = SurelyApart,
     for any θ. (Property 12 from [CTF])

(U4) Apart types do not unify.
     If (unify ξ τ♭) = SurelyApart, then there exists no θ
     such that θ(ξ) = θ(τ). (Property 13 from [CTF])

THEOREM. Completeness w.r.t ~
    If (unify τ₁♭ τ₂♭) = SurelyApart,
    then there exists no proof that (τ₁ ~ τ₂).

PROOF. See appendix of [CTF].


The unification algorithm is used for type family injectivity, as described
in the "Injective Type Families" paper (Haskell'15), called [ITF]. When run
in this mode, it has the following properties.

(I1) If (unify σ τ) = SurelyApart, then σ and τ are not unifiable, even
     after arbitrary type family reductions. Note that σ and τ are
     not flattened here.

(I2) If (unify σ τ) = MaybeApart θ, and if some
     φ exists such that φ(σ) ~ φ(τ), then φ extends θ.


Furthermore, the RULES matching algorithm requires this property,
but only when using this algorithm for matching:

(M1) If (match σ τ) succeeds with θ, then all matchable tyvars
     in σ are bound in θ.

     Property M1 means that we must extend the substitution with,
     say (a ↦ a) when appropriate during matching.
     See also Note [Self-substitution when matching].

(M2) Completeness of matching.
     If θ(σ) = τ, then (match σ τ) = Unifiable φ,
     where θ is an extension of φ.

Sadly, property M2 and I2 conflict. Consider

type family F1 a b where
  F1 Int    Bool   = Char
  F1 Double String = Char

Consider now two matching problems:

P1. match (F1 a Bool) (F1 Int Bool)
P2. match (F1 a Bool) (F1 Double String)

In case P1, we must find (a ↦ Int) to satisfy M2.
In case P2, we must /not/ find (a ↦ Double), in order to satisfy I2. (Note
that the correct mapping for I2 is (a ↦ Int). There is no way to discover
this, but we mustn't map a to anything else!)

We thus must parameterize the algorithm over whether it's being used
for an injectivity check (refrain from looking at non-injective arguments
to type families) or not (do indeed look at those arguments).  This is
implemented  by the um_inj_tf field of UMEnv.

(It's all a question of whether or not to include equation (7) from Fig. 2
of [ITF].)

This extra parameter is a bit fiddly, perhaps, but seemingly less so than
having two separate, almost-identical algorithms.

Note [Self-substitution when matching]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
What should happen when we're *matching* (not unifying) a1 with a1? We
should get a substitution [a1 |-> a1]. A successful match should map all
the template variables (except ones that disappear when expanding synonyms).
But when unifying, we don't want to do this, because we'll then fall into
a loop.

This arrangement affects the code in three places:
 - If we're matching a refined template variable, don't recur. Instead, just
   check for equality. That is, if we know [a |-> Maybe a] and are matching
   (a ~? Maybe Int), we want to just fail.

 - Skip the occurs check when matching. This comes up in two places, because
   matching against variables is handled separately from matching against
   full-on types.

Note that this arrangement was provoked by a real failure, where the same
unique ended up in the template as in the target. (It was a rule firing when
compiling Data.List.NonEmpty.)

Note [Matching coercion variables]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider this:

   type family F a

   data G a where
     MkG :: F a ~ Bool => G a

   type family Foo (x :: G a) :: F a
   type instance Foo MkG = False

We would like that to be accepted. For that to work, we need to introduce
a coercion variable on the left and then use it on the right. Accordingly,
at use sites of Foo, we need to be able to use matching to figure out the
value for the coercion. (See the desugared version:

   axFoo :: [a :: *, c :: F a ~ Bool]. Foo (MkG c) = False |> (sym c)

) We never want this action to happen during *unification* though, when
all bets are off.

Note [Kind coercions in Unify]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We wish to match/unify while ignoring casts. But, we can't just ignore
them completely, or we'll end up with ill-kinded substitutions. For example,
say we're matching `a` with `ty |> co`. If we just drop the cast, we'll
return [a |-> ty], but `a` and `ty` might have different kinds. We can't
just match/unify their kinds, either, because this might gratuitously
fail. After all, `co` is the witness that the kinds are the same -- they
may look nothing alike.

So, we pass a kind coercion to the match/unify worker. This coercion witnesses
the equality between the substed kind of the left-hand type and the substed
kind of the right-hand type. Note that we do not unify kinds at the leaves
(as we did previously). We thus have

Hence: (Unification Kind Invariant)
-----------------------------------
In the call
     unify_ty ty1 ty2 kco
it must be that
     subst(kco) :: subst(kind(ty1)) ~N subst(kind(ty2))
where `subst` is the ambient substitution in the UM monad.  And in the call
     unify_tys tys1 tys2
(which has no kco), after we unify any prefix of tys1,tys2, the kinds of the
head of the remaining tys1,tys2 are identical after substitution.  This
implies, for example, that the kinds of the head of tys1,tys2 are identical
after substitution.

To get this coercion, we first have to match/unify
the kinds before looking at the types. Happily, we need look only one level
up, as all kinds are guaranteed to have kind *.

When we're working with type applications (either TyConApp or AppTy) we
need to worry about establishing INVARIANT, as the kinds of the function
& arguments aren't (necessarily) included in the kind of the result.
When unifying two TyConApps, this is easy, because the two TyCons are
the same. Their kinds are thus the same. As long as we unify left-to-right,
we'll be sure to unify types' kinds before the types themselves. (For example,
think about Proxy :: forall k. k -> *. Unifying the first args matches up
the kinds of the second args.)

For AppTy, we must unify the kinds of the functions, but once these are
unified, we can continue unifying arguments without worrying further about
kinds.

The interface to this module includes both "...Ty" functions and
"...TyKi" functions. The former assume that INVARIANT is already
established, either because the kinds are the same or because the
list of types being passed in are the well-typed arguments to some
type constructor (see two paragraphs above). The latter take a separate
pre-pass over the kinds to establish INVARIANT. Sometimes, it's important
not to take the second pass, as it caused #12442.

We thought, at one point, that this was all unnecessary: why should
casts be in types in the first place? But they are sometimes. In
dependent/should_compile/KindEqualities2, we see, for example the
constraint Num (Int |> (blah ; sym blah)).  We naturally want to find
a dictionary for that constraint, which requires dealing with
coercions in this manner.

Note [Matching in the presence of casts (1)]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When matching, it is crucial that no variables from the template
end up in the range of the matching substitution (obviously!).
When unifying, that's not a constraint; instead we take the fixpoint
of the substitution at the end.

So what should we do with this, when matching?
   unify_ty (tmpl |> co) tgt kco

Previously, wrongly, we pushed 'co' in the (horrid) accumulating
'kco' argument like this:
   unify_ty (tmpl |> co) tgt kco
     = unify_ty tmpl tgt (kco ; co)

But that is obviously wrong because 'co' (from the template) ends
up in 'kco', which in turn ends up in the range of the substitution.

This all came up in #13910.  Because we match tycon arguments
left-to-right, the ambient substitution will already have a matching
substitution for any kinds; so there is an easy fix: just apply
the substitution-so-far to the coercion from the LHS.

Note that

* When matching, the first arg of unify_ty is always the template;
  we never swap round.

* The above argument is distressingly indirect. We seek a
  better way.

* One better way is to ensure that type patterns (the template
  in the matching process) have no casts.  See #14119.

Note [Matching in the presence of casts (2)]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
There is another wrinkle (#17395).  Suppose (T :: forall k. k -> Type)
and we are matching
   tcMatchTy (T k (a::k))  (T j (b::j))

Then we'll match k :-> j, as expected. But then in unify_tys
we invoke
   unify_tys env (a::k) (b::j) (Refl j)

Although we have unified k and j, it's very important that we put
(Refl j), /not/ (Refl k) as the fourth argument to unify_tys.
If we put (Refl k) we'd end up with the substitution
  a :-> b |> Refl k
which is bogus because one of the template variables, k,
appears in the range of the substitution.  Eek.

Similar care is needed in unify_ty_app.


Note [Polykinded tycon applications]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose  T :: forall k. Type -> K
and we are unifying
  ty1:  T @Type         Int       :: Type
  ty2:  T @(Type->Type) Int Int   :: Type

These two TyConApps have the same TyCon at the front but they
(legitimately) have different numbers of arguments.  They
are surelyApart, so we can report that without looking any
further (see #15704).

Note [Unifying type applications]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Unifying type applications is quite subtle, as we found
in #23134 and #22647, when type families are involved.

Suppose
   type family F a :: Type -> Type
   type family G k :: k = r | r -> k

and consider these examples:

* F Int ~ F Char, where F is injective
  Since F is injective, we can reduce this to Int ~ Char,
  therefore SurelyApart.

* F Int ~ F Char, where F is not injective
  Without injectivity, return MaybeApart.

* G Type ~ G (Type -> Type) Int
  Even though G is injective and the arguments to G are different,
  we cannot deduce apartness because the RHS is oversaturated.
  For example, G might be defined as
    G Type = Maybe Int
    G (Type -> Type) = Maybe
  So we return MaybeApart.

* F Int Bool ~ F Int Char       -- SurelyApart (since Bool is apart from Char)
  F Int Bool ~ Maybe a          -- MaybeApart
  F Int Bool ~ a b              -- MaybeApart
  F Int Bool ~ Char -> Bool     -- MaybeApart
  An oversaturated type family can match an application,
  whether it's a TyConApp, AppTy or FunTy. Decompose.

* F Int ~ a b
  We cannot decompose a saturated, or under-saturated
  type family application. We return MaybeApart.

To handle all those conditions, unify_ty goes through
the following checks in sequence, where Fn is a type family
of arity n:

* (C1) Fn x_1 ... x_n ~ Fn y_1 .. y_n
  A saturated application.
  Here we can unify arguments in which Fn is injective.
* (C2) Fn x_1 ... x_n ~ anything, anything ~ Fn x_1 ... x_n
  A saturated type family can match anything - we return MaybeApart.
* (C3) Fn x_1 ... x_m ~ a b, a b ~ Fn x_1 ... x_m where m > n
  An oversaturated type family can be decomposed.
* (C4) Fn x_1 ... x_m ~ anything, anything ~ Fn x_1 ... x_m, where m > n
  If we couldn't decompose in the previous step, we return SurelyApart.

Afterwards, the rest of the code doesn't have to worry about type families.
-}

-------------- unify_ty: the main workhorse -----------

type AmIUnifying = Bool   -- True  <=> Unifying
                          -- False <=> Matching

unify_ty :: UMEnv
         -> Type -> Type  -- Types to be unified and a co
         -> CoercionN     -- A coercion between their kinds
                          -- See Note [Kind coercions in Unify]
         -> UM ()
-- Precondition: see (Unification Kind Invariant)
--
-- See Note [Specification of unification]
-- Respects newtypes, PredTypes
-- See Note [Computing equality on types] in GHC.Core.Type
unify_ty _env (TyConApp tc1 []) (TyConApp tc2 []) _kco
  -- See Note [Comparing nullary type synonyms] in GHC.Core.Type.
  | tc1 == tc2
  = return ()

unify_ty env ty1 ty2 kco
    -- Now handle the cases we can "look through": synonyms and casts.
  | Just ty1' <- coreView ty1 = unify_ty env ty1' ty2 kco
  | Just ty2' <- coreView ty2 = unify_ty env ty1 ty2' kco
  | CastTy ty1' co <- ty1     = if um_unif env
                                then unify_ty env ty1' ty2 (co `mkTransCo` kco)
                                else -- See Note [Matching in the presence of casts (1)]
                                     do { subst <- getSubst env
                                        ; let co' = substCo subst co
                                        ; unify_ty env ty1' ty2 (co' `mkTransCo` kco) }
  | CastTy ty2' co <- ty2     = unify_ty env ty1 ty2' (kco `mkTransCo` mkSymCo co)

unify_ty env (TyVarTy tv1) ty2 kco
  = uVar env tv1 ty2 kco
unify_ty env ty1 (TyVarTy tv2) kco
  | um_unif env  -- If unifying, can swap args
  = uVar (umSwapRn env) tv2 ty1 (mkSymCo kco)

unify_ty env ty1 ty2 _kco

  -- Handle non-oversaturated type families first
  -- See Note [Unifying type applications]
  --
  -- (C1) If we have T x1 ... xn ~ T y1 ... yn, use injectivity information of T
  -- Note that both sides must not be oversaturated
  | Just (tc1, tys1) <- isSatTyFamApp mb_tc_app1
  , Just (tc2, tys2) <- isSatTyFamApp mb_tc_app2
  , tc1 == tc2
  = do { let inj = case tyConInjectivityInfo tc1 of
                          NotInjective -> repeat False
                          Injective bs -> bs

             (inj_tys1, noninj_tys1) = partitionByList inj tys1
             (inj_tys2, noninj_tys2) = partitionByList inj tys2

       ; unify_tys env inj_tys1 inj_tys2
       ; unless (um_inj_tf env) $ -- See (end of) Note [Specification of unification]
         don'tBeSoSure MARTypeFamily $ unify_tys env noninj_tys1 noninj_tys2 }

  | Just _ <- isSatTyFamApp mb_tc_app1  -- (C2) A (not-over-saturated) type-family application
  = maybeApart MARTypeFamily            -- behaves like a type variable; might match

  | Just _ <- isSatTyFamApp mb_tc_app2  -- (C2) A (not-over-saturated) type-family application
                                        -- behaves like a type variable; might unify
                                        -- but doesn't match (as in the TyVarTy case)
  = if um_unif env then maybeApart MARTypeFamily else surelyApart

  -- Handle oversaturated type families.
  --
  -- They can match an application (TyConApp/FunTy/AppTy), this is handled
  -- the same way as in the AppTy case below.
  --
  -- If there is no application, an oversaturated type family can only
  -- match a type variable or a saturated type family,
  -- both of which we handled earlier. So we can say surelyApart.
  | Just (tc1, _) <- mb_tc_app1
  , isTypeFamilyTyCon tc1
  = if | Just (ty1a, ty1b) <- tcSplitAppTyNoView_maybe ty1
       , Just (ty2a, ty2b) <- tcSplitAppTyNoView_maybe ty2
       -> unify_ty_app env ty1a [ty1b] ty2a [ty2b]            -- (C3)
       | otherwise -> surelyApart                             -- (C4)

  | Just (tc2, _) <- mb_tc_app2
  , isTypeFamilyTyCon tc2
  = if | Just (ty1a, ty1b) <- tcSplitAppTyNoView_maybe ty1
       , Just (ty2a, ty2b) <- tcSplitAppTyNoView_maybe ty2
       -> unify_ty_app env ty1a [ty1b] ty2a [ty2b]            -- (C3)
       | otherwise -> surelyApart                             -- (C4)

  -- At this point, neither tc1 nor tc2 can be a type family.
  | Just (tc1, tys1) <- mb_tc_app1
  , Just (tc2, tys2) <- mb_tc_app2
  , tc1 == tc2
  = do { massertPpr (isInjectiveTyCon tc1 Nominal) (ppr tc1)
       ; unify_tys env tys1 tys2
       }

  -- TYPE and CONSTRAINT are not Apart
  -- See Note [Type and Constraint are not apart] in GHC.Builtin.Types.Prim
  -- NB: at this point we know that the two TyCons do not match
  | Just (tc1,_) <- mb_tc_app1, let u1 = tyConUnique tc1
  , Just (tc2,_) <- mb_tc_app2, let u2 = tyConUnique tc2
  , (u1 == tYPETyConKey && u2 == cONSTRAINTTyConKey) ||
    (u2 == tYPETyConKey && u1 == cONSTRAINTTyConKey)
  = maybeApart MARTypeVsConstraint
    -- We don't bother to look inside; wrinkle (W3) in GHC.Builtin.Types.Prim
    -- Note [Type and Constraint are not apart]

  -- The arrow types are not Apart
  -- See Note [Type and Constraint are not apart] in GHC.Builtin.Types.Prim
  --     wrinkle (W2)
  -- NB1: at this point we know that the two TyCons do not match
  -- NB2: In the common FunTy/FunTy case you might wonder if we want to go via
  --      splitTyConApp_maybe.  But yes we do: we need to look at those implied
  --      kind argument in order to satisfy (Unification Kind Invariant)
  | FunTy {} <- ty1
  , FunTy {} <- ty2
  = maybeApart MARTypeVsConstraint
    -- We don't bother to look inside; wrinkle (W3) in GHC.Builtin.Types.Prim
    -- Note [Type and Constraint are not apart]

  where
    mb_tc_app1 = splitTyConApp_maybe ty1
    mb_tc_app2 = splitTyConApp_maybe ty2

        -- Applications need a bit of care!
        -- They can match FunTy and TyConApp, so use splitAppTy_maybe
        -- NB: we've already dealt with type variables,
        -- so if one type is an App the other one jolly well better be too
unify_ty env (AppTy ty1a ty1b) ty2 _kco
  | Just (ty2a, ty2b) <- tcSplitAppTyNoView_maybe ty2
  = unify_ty_app env ty1a [ty1b] ty2a [ty2b]

unify_ty env ty1 (AppTy ty2a ty2b) _kco
  | Just (ty1a, ty1b) <- tcSplitAppTyNoView_maybe ty1
  = unify_ty_app env ty1a [ty1b] ty2a [ty2b]

unify_ty _ (LitTy x) (LitTy y) _kco | x == y = return ()

unify_ty env (ForAllTy (Bndr tv1 _) ty1) (ForAllTy (Bndr tv2 _) ty2) kco
  = do { unify_ty env (varType tv1) (varType tv2) (mkNomReflCo liftedTypeKind)
       ; let env' = umRnBndr2 env tv1 tv2
       ; unify_ty env' ty1 ty2 kco }

-- See Note [Matching coercion variables]
unify_ty env (CoercionTy co1) (CoercionTy co2) kco
  = do { c_subst <- getCvSubstEnv
       ; case co1 of
           CoVarCo cv
             | not (um_unif env)
             , not (cv `elemVarEnv` c_subst)
             , let (_, co_l, co_r) = decomposeFunCo kco
                     -- Because the coercion is used in a type, it should be safe to
                     -- ignore the multiplicity coercion.
                      -- cv :: t1 ~ t2
                      -- co2 :: s1 ~ s2
                      -- co_l :: t1 ~ s1
                      -- co_r :: t2 ~ s2
                   rhs_co = co_l `mkTransCo` co2 `mkTransCo` mkSymCo co_r
             , BindMe <- tvBindFlag env cv (CoercionTy rhs_co)
             -> do { checkRnEnv env (tyCoVarsOfCo co2)
                   ; extendCvEnv cv rhs_co }
           _ -> return () }

unify_ty _ _ _ _ = surelyApart

unify_ty_app :: UMEnv -> Type -> [Type] -> Type -> [Type] -> UM ()
unify_ty_app env ty1 ty1args ty2 ty2args
  | Just (ty1', ty1a) <- splitAppTyNoView_maybe ty1
  , Just (ty2', ty2a) <- splitAppTyNoView_maybe ty2
  = unify_ty_app env ty1' (ty1a : ty1args) ty2' (ty2a : ty2args)

  | otherwise
  = do { let ki1 = typeKind ty1
             ki2 = typeKind ty2
           -- See Note [Kind coercions in Unify]
       ; unify_ty  env ki1 ki2 (mkNomReflCo liftedTypeKind)
       ; unify_ty  env ty1 ty2 (mkNomReflCo ki2)
                 -- Very important: 'ki2' not 'ki1'
                 -- See Note [Matching in the presence of casts (2)]
       ; unify_tys env ty1args ty2args }

unify_tys :: UMEnv -> [Type] -> [Type] -> UM ()
-- Precondition: see (Unification Kind Invariant)
unify_tys env orig_xs orig_ys
  = go orig_xs orig_ys
  where
    go []     []     = return ()
    go (x:xs) (y:ys)
      -- See Note [Kind coercions in Unify]
      = do { unify_ty env x y (mkNomReflCo $ typeKind y)
                 -- Very important: 'y' not 'x'
                 -- See Note [Matching in the presence of casts (2)]
           ; go xs ys }
    go _ _ = surelyApart
      -- Possibly different saturations of a polykinded tycon
      -- See Note [Polykinded tycon applications]

isSatTyFamApp :: Maybe (TyCon, [Type]) -> Maybe (TyCon, [Type])
-- Return the argument if we have a saturated type family application
-- If it is /over/ saturated then we return False.  E.g.
--     unify_ty (F a b) (c d)    where F has arity 1
-- we definitely want to decompose that type application! (#22647)
isSatTyFamApp tapp@(Just (tc, tys))
  |  isTypeFamilyTyCon tc
  && not (tys `lengthExceeds` tyConArity tc)  -- Not over-saturated
  = tapp
isSatTyFamApp _ = Nothing

---------------------------------
uVar :: UMEnv
     -> InTyVar         -- Variable to be unified
     -> Type            -- with this Type
     -> Coercion        -- :: kind tv ~N kind ty
     -> UM ()

uVar env tv1 ty kco
 = do { -- Apply the ambient renaming
        let tv1' = umRnOccL env tv1

        -- Check to see whether tv1 is refined by the substitution
      ; subst <- getTvSubstEnv
      ; case (lookupVarEnv subst tv1') of
          Just ty' | um_unif env                -- Unifying, so call
                   -> unify_ty env ty' ty kco   -- back into unify
                   | otherwise
                   -> -- Matching, we don't want to just recur here.
                      -- this is because the range of the subst is the target
                      -- type, not the template type. So, just check for
                      -- normal type equality.
                      unless ((ty' `mkCastTy` kco) `tcEqType` ty) $
                        surelyApart
                      -- NB: it's important to use `tcEqType` instead of `eqType` here,
                      -- otherwise we might not reject a substitution
                      -- which unifies `Type` with `Constraint`, e.g.
                      -- a call to tc_unify_tys with arguments
                      --
                      --   tys1 = [k,k]
                      --   tys2 = [Type, Constraint]
                      --
                      -- See test cases: T11715b, T20521.
          Nothing  -> uUnrefined env tv1' ty ty kco } -- No, continue

uUnrefined :: UMEnv
           -> OutTyVar          -- variable to be unified
           -> Type              -- with this Type
           -> Type              -- (version w/ expanded synonyms)
           -> Coercion          -- :: kind tv ~N kind ty
           -> UM ()

-- We know that tv1 isn't refined

uUnrefined env tv1' ty2 ty2' kco
  | Just ty2'' <- coreView ty2'
  = uUnrefined env tv1' ty2 ty2'' kco    -- Unwrap synonyms
                -- This is essential, in case we have
                --      type Foo a = a
                -- and then unify a ~ Foo a

  | TyVarTy tv2 <- ty2'
  = do { let tv2' = umRnOccR env tv2
       ; unless (tv1' == tv2' && um_unif env) $ do
           -- If we are unifying a ~ a, just return immediately
           -- Do not extend the substitution
           -- See Note [Self-substitution when matching]

          -- Check to see whether tv2 is refined
       { subst <- getTvSubstEnv
       ; case lookupVarEnv subst tv2 of
         {  Just ty' | um_unif env -> uUnrefined env tv1' ty' ty' kco
         ;  _ ->

    do {   -- So both are unrefined
           -- Bind one or the other, depending on which is bindable
       ; let rhs1 = ty2 `mkCastTy` mkSymCo kco
             rhs2 = ty1 `mkCastTy` kco
             b1  = tvBindFlag env tv1' rhs1
             b2  = tvBindFlag env tv2' rhs2
             ty1 = mkTyVarTy tv1'
       ; case (b1, b2) of
           (BindMe, _) -> bindTv env tv1' rhs1
           (_, BindMe) | um_unif env
                       -> bindTv (umSwapRn env) tv2 rhs2

           _ | tv1' == tv2' -> return ()
             -- How could this happen? If we're only matching and if
             -- we're comparing forall-bound variables.

           _ -> surelyApart
  }}}}

uUnrefined env tv1' ty2 _ kco -- ty2 is not a type variable
  = case tvBindFlag env tv1' rhs of
      Apart  -> surelyApart
      BindMe -> bindTv env tv1' rhs
  where
    rhs = ty2 `mkCastTy` mkSymCo kco

bindTv :: UMEnv -> OutTyVar -> Type -> UM ()
-- OK, so we want to extend the substitution with tv := ty
-- But first, we must do a couple of checks
bindTv env tv1 ty2
  = do  { let free_tvs2 = tyCoVarsOfType ty2

        -- Make sure tys mentions no local variables
        -- E.g.  (forall a. b) ~ (forall a. [a])
        -- We should not unify b := [a]!
        ; checkRnEnv env free_tvs2

        -- Occurs check, see Note [Fine-grained unification]
        -- Make sure you include 'kco' (which ty2 does) #14846
        ; occurs <- occursCheck env tv1 free_tvs2

        ; if occurs then maybeApart MARInfinite
                    else extendTvEnv tv1 ty2 }

occursCheck :: UMEnv -> TyVar -> VarSet -> UM Bool
occursCheck env tv free_tvs
  | um_unif env
  = do { tsubst <- getTvSubstEnv
       ; return (tv `elemVarSet` niSubstTvSet tsubst free_tvs) }

  | otherwise      -- Matching; no occurs check
  = return False   -- See Note [Self-substitution when matching]

{-
%************************************************************************
%*                                                                      *
                Binding decisions
*                                                                      *
************************************************************************
-}

data BindFlag
  = BindMe      -- ^ A regular type variable

  | Apart       -- ^ Declare that this type variable is /apart/ from the
                -- type provided. That is, the type variable will never
                -- be instantiated to that type.
                -- See also Note [Binding when looking up instances]
                -- in GHC.Core.InstEnv.
  deriving Eq
-- NB: It would be conceivable to have an analogue to MaybeApart here,
-- but there is not yet a need.

{-
************************************************************************
*                                                                      *
                Unification monad
*                                                                      *
************************************************************************
-}

data UMEnv
  = UMEnv { um_unif :: AmIUnifying

          , um_inj_tf :: Bool
            -- Checking for injectivity?
            -- See (end of) Note [Specification of unification]

          , um_rn_env :: RnEnv2
            -- Renaming InTyVars to OutTyVars; this eliminates
            -- shadowing, and lines up matching foralls on the left
            -- and right

          , um_skols :: TyVarSet
            -- OutTyVars bound by a forall in this unification;
            -- Do not bind these in the substitution!
            -- See the function tvBindFlag

          , um_bind_fun :: BindFun
            -- User-supplied BindFlag function,
            -- for variables not in um_skols
          }

data UMState = UMState
                   { um_tv_env   :: TvSubstEnv
                   , um_cv_env   :: CvSubstEnv }

newtype UM a
  = UM' { unUM :: UMState -> UnifyResultM (UMState, a) }
    -- See Note [The one-shot state monad trick] in GHC.Utils.Monad

pattern UM :: (UMState -> UnifyResultM (UMState, a)) -> UM a
-- See Note [The one-shot state monad trick] in GHC.Utils.Monad
pattern UM m <- UM' m
  where
    UM m = UM' (oneShot m)
{-# COMPLETE UM #-}

instance Functor UM where
  fmap f (UM m) = UM (\s -> fmap (\(s', v) -> (s', f v)) (m s))

instance Applicative UM where
      pure a = UM (\s -> pure (s, a))
      (<*>)  = ap

instance Monad UM where
  {-# INLINE (>>=) #-}
  -- See Note [INLINE pragmas and (>>)] in GHC.Utils.Monad
  m >>= k  = UM (\state ->
                  do { (state', v) <- unUM m state
                     ; unUM (k v) state' })

instance MonadFail UM where
    fail _   = UM (\_ -> SurelyApart) -- failed pattern match

initUM :: TvSubstEnv  -- subst to extend
       -> CvSubstEnv
       -> UM a -> UnifyResultM a
initUM subst_env cv_subst_env um
  = case unUM um state of
      Unifiable (_, subst)    -> Unifiable subst
      MaybeApart r (_, subst) -> MaybeApart r subst
      SurelyApart             -> SurelyApart
  where
    state = UMState { um_tv_env = subst_env
                    , um_cv_env = cv_subst_env }

tvBindFlag :: UMEnv -> OutTyVar -> Type -> BindFlag
tvBindFlag env tv rhs
  | tv `elemVarSet` um_skols env = Apart
  | otherwise                    = um_bind_fun env tv rhs

getTvSubstEnv :: UM TvSubstEnv
getTvSubstEnv = UM $ \state -> Unifiable (state, um_tv_env state)

getCvSubstEnv :: UM CvSubstEnv
getCvSubstEnv = UM $ \state -> Unifiable (state, um_cv_env state)

getSubst :: UMEnv -> UM Subst
getSubst env = do { tv_env <- getTvSubstEnv
                  ; cv_env <- getCvSubstEnv
                  ; let in_scope = rnInScopeSet (um_rn_env env)
                  ; return (mkSubst in_scope tv_env cv_env emptyIdSubstEnv) }

extendTvEnv :: TyVar -> Type -> UM ()
extendTvEnv tv ty = UM $ \state ->
  Unifiable (state { um_tv_env = extendVarEnv (um_tv_env state) tv ty }, ())

extendCvEnv :: CoVar -> Coercion -> UM ()
extendCvEnv cv co = UM $ \state ->
  Unifiable (state { um_cv_env = extendVarEnv (um_cv_env state) cv co }, ())

umRnBndr2 :: UMEnv -> TyCoVar -> TyCoVar -> UMEnv
umRnBndr2 env v1 v2
  = env { um_rn_env = rn_env', um_skols = um_skols env `extendVarSet` v' }
  where
    (rn_env', v') = rnBndr2_var (um_rn_env env) v1 v2

checkRnEnv :: UMEnv -> VarSet -> UM ()
checkRnEnv env varset
  | isEmptyVarSet skol_vars           = return ()
  | varset `disjointVarSet` skol_vars = return ()
  | otherwise                         = surelyApart
  where
    skol_vars = um_skols env
    -- NB: That isEmptyVarSet guard is a critical optimization;
    -- it means we don't have to calculate the free vars of
    -- the type, often saving quite a bit of allocation.

-- | Converts any SurelyApart to a MaybeApart
don'tBeSoSure :: MaybeApartReason -> UM () -> UM ()
don'tBeSoSure r um = UM $ \ state ->
  case unUM um state of
    SurelyApart -> MaybeApart r (state, ())
    other       -> other

umRnOccL :: UMEnv -> TyVar -> TyVar
umRnOccL env v = rnOccL (um_rn_env env) v

umRnOccR :: UMEnv -> TyVar -> TyVar
umRnOccR env v = rnOccR (um_rn_env env) v

umSwapRn :: UMEnv -> UMEnv
umSwapRn env = env { um_rn_env = rnSwap (um_rn_env env) }

maybeApart :: MaybeApartReason -> UM ()
maybeApart r = UM (\state -> MaybeApart r (state, ()))

surelyApart :: UM a
surelyApart = UM (\_ -> SurelyApart)

{-
%************************************************************************
%*                                                                      *
            Matching a (lifted) type against a coercion
%*                                                                      *
%************************************************************************

This section defines essentially an inverse to liftCoSubst. It is defined
here to avoid a dependency from Coercion on this module.

-}

data MatchEnv = ME { me_tmpls :: TyVarSet
                   , me_env   :: RnEnv2 }

-- | 'liftCoMatch' is sort of inverse to 'liftCoSubst'.  In particular, if
--   @liftCoMatch vars ty co == Just s@, then @liftCoSubst s ty == co@,
--   where @==@ there means that the result of 'liftCoSubst' has the same
--   type as the original co; but may be different under the hood.
--   That is, it matches a type against a coercion of the same
--   "shape", and returns a lifting substitution which could have been
--   used to produce the given coercion from the given type.
--   Note that this function is incomplete -- it might return Nothing
--   when there does indeed exist a possible lifting context.
--
-- This function is incomplete in that it doesn't respect the equality
-- in `eqType`. That is, it's possible that this will succeed for t1 and
-- fail for t2, even when t1 `eqType` t2. That's because it depends on
-- there being a very similar structure between the type and the coercion.
-- This incompleteness shouldn't be all that surprising, especially because
-- it depends on the structure of the coercion, which is a silly thing to do.
--
-- The lifting context produced doesn't have to be exacting in the roles
-- of the mappings. This is because any use of the lifting context will
-- also require a desired role. Thus, this algorithm prefers mapping to
-- nominal coercions where it can do so.
liftCoMatch :: TyCoVarSet -> Type -> Coercion -> Maybe LiftingContext
liftCoMatch tmpls ty co
  = do { cenv1 <- ty_co_match menv emptyVarEnv ki ki_co ki_ki_co ki_ki_co
       ; cenv2 <- ty_co_match menv cenv1       ty co
                              (mkNomReflCo co_lkind) (mkNomReflCo co_rkind)
       ; return (LC (mkEmptySubst in_scope) cenv2) }
  where
    menv     = ME { me_tmpls = tmpls, me_env = mkRnEnv2 in_scope }
    in_scope = mkInScopeSet (tmpls `unionVarSet` tyCoVarsOfCo co)
    -- Like tcMatchTy, assume all the interesting variables
    -- in ty are in tmpls

    ki       = typeKind ty
    ki_co    = promoteCoercion co
    ki_ki_co = mkNomReflCo liftedTypeKind

    Pair co_lkind co_rkind = coercionKind ki_co

-- | 'ty_co_match' does all the actual work for 'liftCoMatch'.
ty_co_match :: MatchEnv   -- ^ ambient helpful info
            -> LiftCoEnv  -- ^ incoming subst
            -> Type       -- ^ ty, type to match
            -> Coercion   -- ^ co :: lty ~r rty, coercion to match against
            -> Coercion   -- ^ :: kind(lsubst(ty)) ~N kind(lty)
            -> Coercion   -- ^ :: kind(rsubst(ty)) ~N kind(rty)
            -> Maybe LiftCoEnv
   -- ^ Just env ==> liftCoSubst Nominal env ty == co, modulo roles.
   -- Also: Just env ==> lsubst(ty) == lty and rsubst(ty) == rty,
   -- where lsubst = lcSubstLeft(env) and rsubst = lcSubstRight(env)
ty_co_match menv subst ty co lkco rkco
  | Just ty' <- coreView ty = ty_co_match menv subst ty' co lkco rkco

  -- handle Refl case:
  | tyCoVarsOfType ty `isNotInDomainOf` subst
  , Just (ty', _) <- isReflCo_maybe co
  , ty `eqType` ty'
    -- Why `eqType` and not `tcEqType`? Because this function is only used
    -- during coercion optimisation, after type-checking has finished.
  = Just subst

  where
    isNotInDomainOf :: VarSet -> VarEnv a -> Bool
    isNotInDomainOf set env
      = noneSet (\v -> elemVarEnv v env) set

    noneSet :: (Var -> Bool) -> VarSet -> Bool
    noneSet f = allVarSet (not . f)

ty_co_match menv subst ty co lkco rkco
  | CastTy ty' co' <- ty
     -- See Note [Matching in the presence of casts (1)]
  = let empty_subst  = mkEmptySubst (rnInScopeSet (me_env menv))
        substed_co_l = substCo (liftEnvSubstLeft empty_subst subst)  co'
        substed_co_r = substCo (liftEnvSubstRight empty_subst subst) co'
    in
    ty_co_match menv subst ty' co (substed_co_l `mkTransCo` lkco)
                                  (substed_co_r `mkTransCo` rkco)

  | SymCo co' <- co
  = swapLiftCoEnv <$> ty_co_match menv (swapLiftCoEnv subst) ty co' rkco lkco

  -- Match a type variable against a non-refl coercion
ty_co_match menv subst (TyVarTy tv1) co lkco rkco
  | Just co1' <- lookupVarEnv subst tv1' -- tv1' is already bound to co1
  = if eqCoercionX (nukeRnEnvL rn_env) co1' co
    then Just subst
    else Nothing       -- no match since tv1 matches two different coercions

  | tv1' `elemVarSet` me_tmpls menv           -- tv1' is a template var
  = if any (inRnEnvR rn_env) (tyCoVarsOfCoList co)
    then Nothing      -- occurs check failed
    else Just $ extendVarEnv subst tv1' $
                castCoercionKind co (mkSymCo lkco) (mkSymCo rkco)

  | otherwise
  = Nothing

  where
    rn_env = me_env menv
    tv1' = rnOccL rn_env tv1

  -- just look through SubCo's. We don't really care about roles here.
ty_co_match menv subst ty (SubCo co) lkco rkco
  = ty_co_match menv subst ty co lkco rkco

ty_co_match menv subst (AppTy ty1a ty1b) co _lkco _rkco
  | Just (co2, arg2) <- splitAppCo_maybe co     -- c.f. Unify.match on AppTy
  = ty_co_match_app menv subst ty1a [ty1b] co2 [arg2]
ty_co_match menv subst ty1 (AppCo co2 arg2) _lkco _rkco
  | Just (ty1a, ty1b) <- splitAppTyNoView_maybe ty1
       -- yes, the one from Type, not TcType; this is for coercion optimization
  = ty_co_match_app menv subst ty1a [ty1b] co2 [arg2]

ty_co_match menv subst (TyConApp tc1 tys) (TyConAppCo _ tc2 cos) _lkco _rkco
  = ty_co_match_tc menv subst tc1 tys tc2 cos

ty_co_match menv subst (FunTy { ft_mult = w, ft_arg = ty1, ft_res = ty2 })
            (FunCo { fco_mult = co_w, fco_arg = co1, fco_res = co2 }) _lkco _rkco
  = ty_co_match_args menv subst [w,    rep1,    rep2,    ty1, ty2]
                                [co_w, co1_rep, co2_rep, co1, co2]
  where
     rep1    = getRuntimeRep ty1
     rep2    = getRuntimeRep ty2
     co1_rep = mkRuntimeRepCo co1
     co2_rep = mkRuntimeRepCo co2
    -- NB: we include the RuntimeRep arguments in the matching;
    --     not doing so caused #21205.

ty_co_match menv subst (ForAllTy (Bndr tv1 _) ty1)
                       (ForAllCo tv2 kind_co2 co2)
                       lkco rkco
  | isTyVar tv1 && isTyVar tv2
  = do { subst1 <- ty_co_match menv subst (tyVarKind tv1) kind_co2
                               ki_ki_co ki_ki_co
       ; let rn_env0 = me_env menv
             rn_env1 = rnBndr2 rn_env0 tv1 tv2
             menv'   = menv { me_env = rn_env1 }
       ; ty_co_match menv' subst1 ty1 co2 lkco rkco }
  where
    ki_ki_co = mkNomReflCo liftedTypeKind

-- ty_co_match menv subst (ForAllTy (Bndr cv1 _) ty1)
--                        (ForAllCo cv2 kind_co2 co2)
--                        lkco rkco
--   | isCoVar cv1 && isCoVar cv2
--   We seems not to have enough information for this case
--   1. Given:
--        cv1      :: (s1 :: k1) ~r (s2 :: k2)
--        kind_co2 :: (s1' ~ s2') ~N (t1 ~ t2)
--        eta1      = mkSelCo (SelTyCon 2 role) (downgradeRole r Nominal kind_co2)
--                 :: s1' ~ t1
--        eta2      = mkSelCo (SelTyCon 3 role) (downgradeRole r Nominal kind_co2)
--                 :: s2' ~ t2
--      Wanted:
--        subst1 <- ty_co_match menv subst  s1 eta1 kco1 kco2
--        subst2 <- ty_co_match menv subst1 s2 eta2 kco3 kco4
--      Question: How do we get kcoi?
--   2. Given:
--        lkco :: <*>    -- See Note [Weird typing rule for ForAllTy] in GHC.Core.TyCo.Rep
--        rkco :: <*>
--      Wanted:
--        ty_co_match menv' subst2 ty1 co2 lkco' rkco'
--      Question: How do we get lkco' and rkco'?

ty_co_match _ subst (CoercionTy {}) _ _ _
  = Just subst -- don't inspect coercions

ty_co_match menv subst ty (GRefl r t (MCo co)) lkco rkco
  =  ty_co_match menv subst ty (GRefl r t MRefl) lkco (rkco `mkTransCo` mkSymCo co)

ty_co_match menv subst ty co1 lkco rkco
  | Just (CastTy t co, r) <- isReflCo_maybe co1
  -- In @pushRefl@, pushing reflexive coercion inside CastTy will give us
  -- t |> co ~ t ; <t> ; t ~ t |> co
  -- But transitive coercions are not helpful. Therefore we deal
  -- with it here: we do recursion on the smaller reflexive coercion,
  -- while propagating the correct kind coercions.
  = let kco' = mkSymCo co
    in ty_co_match menv subst ty (mkReflCo r t) (lkco `mkTransCo` kco')
                                                (rkco `mkTransCo` kco')

ty_co_match menv subst ty co lkco rkco
  | Just co' <- pushRefl co = ty_co_match menv subst ty co' lkco rkco
  | otherwise               = Nothing

ty_co_match_tc :: MatchEnv -> LiftCoEnv
               -> TyCon -> [Type]
               -> TyCon -> [Coercion]
               -> Maybe LiftCoEnv
ty_co_match_tc menv subst tc1 tys1 tc2 cos2
  = do { guard (tc1 == tc2)
       ; ty_co_match_args menv subst tys1 cos2 }

ty_co_match_app :: MatchEnv -> LiftCoEnv
                -> Type -> [Type] -> Coercion -> [Coercion]
                -> Maybe LiftCoEnv
ty_co_match_app menv subst ty1 ty1args co2 co2args
  | Just (ty1', ty1a) <- splitAppTyNoView_maybe ty1
  , Just (co2', co2a) <- splitAppCo_maybe co2
  = ty_co_match_app menv subst ty1' (ty1a : ty1args) co2' (co2a : co2args)

  | otherwise
  = do { subst1 <- ty_co_match menv subst ki1 ki2 ki_ki_co ki_ki_co
       ; let Pair lkco rkco = mkNomReflCo <$> coercionKind ki2
       ; subst2 <- ty_co_match menv subst1 ty1 co2 lkco rkco
       ; ty_co_match_args menv subst2 ty1args co2args }
  where
    ki1 = typeKind ty1
    ki2 = promoteCoercion co2
    ki_ki_co = mkNomReflCo liftedTypeKind

ty_co_match_args :: MatchEnv -> LiftCoEnv -> [Type] -> [Coercion]
                 -> Maybe LiftCoEnv
ty_co_match_args menv subst (ty:tys) (arg:args)
  = do { let Pair lty rty = coercionKind arg
             lkco = mkNomReflCo (typeKind lty)
             rkco = mkNomReflCo (typeKind rty)
       ; subst' <- ty_co_match menv subst ty arg lkco rkco
       ; ty_co_match_args menv subst' tys args }
ty_co_match_args _    subst []       [] = Just subst
ty_co_match_args _    _     _        _  = Nothing

pushRefl :: Coercion -> Maybe Coercion
pushRefl co =
  case (isReflCo_maybe co) of
    Just (AppTy ty1 ty2, Nominal)
      -> Just (AppCo (mkReflCo Nominal ty1) (mkNomReflCo ty2))
    Just (FunTy af w ty1 ty2, r)
      ->  Just (FunCo r af af (mkReflCo r w) (mkReflCo r ty1) (mkReflCo r ty2))
    Just (TyConApp tc tys, r)
      -> Just (TyConAppCo r tc (zipWith mkReflCo (tyConRoleListX r tc) tys))
    Just (ForAllTy (Bndr tv _) ty, r)
      -> Just (ForAllCo tv (mkNomReflCo (varType tv)) (mkReflCo r ty))
    -- NB: NoRefl variant. Otherwise, we get a loop!
    _ -> Nothing

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

Note [Flattening type-family applications when matching instances]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
As described in "Closed type families with overlapping equations"
http://research.microsoft.com/en-us/um/people/simonpj/papers/ext-f/axioms-extended.pdf
we need to flatten core types before unifying them, when checking for "surely-apart"
against earlier equations of a closed type family.
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).

Here is a nice example of why it's all necessary:

  type family F a b where
    F Int Bool = Char
    F a   b    = Double
  type family G a         -- open, no instances

How do we reduce (F (G Float) (G Float))? The first equation clearly doesn't match,
while the second equation does. But, before reducing, we must make sure that the
target can never become (F Int Bool). Well, no matter what G Float becomes, it
certainly won't become *both* Int and Bool, so indeed we're safe reducing
(F (G Float) (G Float)) to Double.

This is necessary not only to get more reductions (which we might be
willing to give up on), but for substitutivity. If we have (F x x), we
can see that (F x x) can reduce to Double. So, it had better be the
case that (F blah blah) can reduce to Double, no matter what (blah)
is!  Flattening as done below ensures this.

We also use this flattening operation to check for class instances.
If we have
  instance C (Maybe b)
  instance {-# OVERLAPPING #-} C (Maybe Bool)
  [W] C (Maybe (F a))
we want to know that the second instance might match later. So we
flatten the (F a) in the target before trying to unify with instances.
(This is done in GHC.Core.InstEnv.lookupInstEnv'.)

The algorithm works by building up a TypeMap TyVar, mapping
type family applications to fresh variables. This mapping must
be threaded through all the function calls, as any entry in
the mapping must be propagated to all future nodes in the tree.

The algorithm also must track the set of in-scope variables, in
order to make fresh variables as it flattens. (We are far from a
source of fresh Uniques.) See Wrinkle 2, below.

There are wrinkles, of course:

1. The flattening algorithm must account for the possibility
   of inner `forall`s. (A `forall` seen here can happen only
   because of impredicativity. However, the flattening operation
   is an algorithm in Core, which is impredicative.)
   Suppose we have (forall b. F b) -> (forall b. F b). Of course,
   those two bs are entirely unrelated, and so we should certainly
   not flatten the two calls F b to the same variable. Instead, they
   must be treated separately. We thus carry a substitution that
   freshens variables; we must apply this substitution (in
   `coreFlattenTyFamApp`) before looking up an application in the environment.
   Note that the range of the substitution contains only TyVars, never anything
   else.

   For the sake of efficiency, we only apply this substitution when absolutely
   necessary. Namely:

   * We do not perform the substitution at all if it is empty.
   * We only need to worry about the arguments of a type family that are within
     the arity of said type family, so we can get away with not applying the
     substitution to any oversaturated type family arguments.
   * Importantly, we do /not/ achieve this substitution by recursively
     flattening the arguments, as this would be wrong. Consider `F (G a)`,
     where F and G are type families. We might decide that `F (G a)` flattens
     to `beta`. Later, the substitution is non-empty (but does not map `a`) and
     so we flatten `G a` to `gamma` and try to flatten `F gamma`. Of course,
     `F gamma` is unknown, and so we flatten it to `delta`, but it really
     should have been `beta`! Argh!

     Moral of the story: instead of flattening the arguments, just substitute
     them directly.

2. There are two different reasons we might add a variable
   to the in-scope set as we work:

     A. We have just invented a new flattening variable.
     B. We have entered a `forall`.

   Annoying here is that in-scope variable source (A) must be
   threaded through the calls. For example, consider (F b -> forall c. F c).
   Suppose that, when flattening F b, we invent a fresh variable c.
   Now, when we encounter (forall c. F c), we need to know c is already in
   scope so that we locally rename c to c'. However, if we don't thread through
   the in-scope set from one argument of (->) to the other, we won't know this
   and might get very confused.

   In contrast, source (B) increases only as we go deeper, as in-scope sets
   normally do. However, even here we must be careful. The TypeMap TyVar that
   contains mappings from type family applications to freshened variables will
   be threaded through both sides of (forall b. F b) -> (forall b. F b). We
   thus must make sure that the two `b`s don't get renamed to the same b1. (If
   they did, then looking up `F b1` would yield the same flatten var for
   each.) So, even though `forall`-bound variables should really be in the
   in-scope set only when they are in scope, we retain these variables even
   outside of their scope. This ensures that, if we encounter a fresh
   `forall`-bound b, we will rename it to b2, not b1. Note that keeping a
   larger in-scope set than strictly necessary is always OK, as in-scope sets
   are only ever used to avoid collisions.

   Sadly, the freshening substitution described in (1) really mustn't bind
   variables outside of their scope: note that its domain is the *unrenamed*
   variables. This means that the substitution gets "pushed down" (like a
   reader monad) while the in-scope set gets threaded (like a state monad).
   Because a Subst contains its own in-scope set, we don't carry a Subst;
   instead, we just carry a TvSubstEnv down, tying it to the InScopeSet
   traveling separately as necessary.

3. Consider `F ty_1 ... ty_n`, where F is a type family with arity k:

     type family F ty_1 ... ty_k :: res_k

   It's tempting to just flatten `F ty_1 ... ty_n` to `alpha`, where alpha is a
   flattening skolem. But we must instead flatten it to
   `alpha ty_(k+1) ... ty_n`—that is, by only flattening up to the arity of the
   type family.

   Why is this better? Consider the following concrete example from #16995:

     type family Param :: Type -> Type

     type family LookupParam (a :: Type) :: Type where
       LookupParam (f Char) = Bool
       LookupParam x        = Int

     foo :: LookupParam (Param ())
     foo = 42

   In order for `foo` to typecheck, `LookupParam (Param ())` must reduce to
   `Int`. But if we flatten `Param ()` to `alpha`, then GHC can't be sure if
   `alpha` is apart from `f Char`, so it won't fall through to the second
   equation. But since the `Param` type family has arity 0, we can instead
   flatten `Param ()` to `alpha ()`, about which GHC knows with confidence is
   apart from `f Char`, permitting the second equation to be reached.

   Not only does this allow more programs to be accepted, it's also important
   for correctness. Not doing this was the root cause of the Core Lint error
   in #16995.

flattenTys is defined here because of module dependencies.
-}

data FlattenEnv
  = FlattenEnv { fe_type_map :: TypeMap (TyVar, TyCon, [Type])
                 -- domain: exactly-saturated type family applications
                 -- range: (fresh variable, type family tycon, args)
               , fe_in_scope :: InScopeSet }
                 -- See Note [Flattening type-family applications when matching instances]

emptyFlattenEnv :: InScopeSet -> FlattenEnv
emptyFlattenEnv in_scope
  = FlattenEnv { fe_type_map = emptyTypeMap
               , fe_in_scope = in_scope }

updateInScopeSet :: FlattenEnv -> (InScopeSet -> InScopeSet) -> FlattenEnv
updateInScopeSet env upd = env { fe_in_scope = upd (fe_in_scope env) }

flattenTys :: InScopeSet -> [Type] -> [Type]
-- See Note [Flattening type-family applications when matching instances]
flattenTys in_scope tys = fst (flattenTysX in_scope tys)

flattenTysX :: InScopeSet -> [Type] -> ([Type], TyVarEnv (TyCon, [Type]))
-- See Note [Flattening type-family applications when matching instances]
-- NB: the returned types mention the fresh type variables
--     in the domain of the returned env, whose range includes
--     the original type family applications. Building a substitution
--     from this information and applying it would yield the original
--     types -- almost. The problem is that the original type might
--     have something like (forall b. F a b); the returned environment
--     can't really sensibly refer to that b. So it may include a locally-
--     bound tyvar in its range. Currently, the only usage of this env't
--     checks whether there are any meta-variables in it
--     (in GHC.Tc.Solver.Monad.mightEqualLater), so this is all OK.
flattenTysX in_scope tys
  = let (env, result) = coreFlattenTys emptyTvSubstEnv (emptyFlattenEnv in_scope) tys in
    (result, build_env (fe_type_map env))
  where
    build_env :: TypeMap (TyVar, TyCon, [Type]) -> TyVarEnv (TyCon, [Type])
    build_env env_in
      = foldTM (\(tv, tc, tys) env_out -> extendVarEnv env_out tv (tc, tys))
               env_in emptyVarEnv

coreFlattenTys :: TvSubstEnv -> FlattenEnv
               -> [Type] -> (FlattenEnv, [Type])
coreFlattenTys subst = mapAccumL (coreFlattenTy subst)

coreFlattenTy :: TvSubstEnv -> FlattenEnv
              -> Type -> (FlattenEnv, Type)
coreFlattenTy subst = go
  where
    go env ty | Just ty' <- coreView ty = go env ty'

    go env (TyVarTy tv)
      | Just ty <- lookupVarEnv subst tv = (env, ty)
      | otherwise                        = let (env', ki) = go env (tyVarKind tv) in
                                           (env', mkTyVarTy $ setTyVarKind tv ki)
    go env (AppTy ty1 ty2) = let (env1, ty1') = go env  ty1
                                 (env2, ty2') = go env1 ty2 in
                             (env2, AppTy ty1' ty2')
    go env (TyConApp tc tys)
         -- NB: Don't just check if isFamilyTyCon: this catches *data* families,
         -- which are generative and thus can be preserved during flattening
      | not (isGenerativeTyCon tc Nominal)
      = coreFlattenTyFamApp subst env tc tys

      | otherwise
      = let (env', tys') = coreFlattenTys subst env tys in
        (env', mkTyConApp tc tys')

    go env ty@(FunTy { ft_mult = mult, ft_arg = ty1, ft_res = ty2 })
      = let (env1, ty1') = go env  ty1
            (env2, ty2') = go env1 ty2
            (env3, mult') = go env2 mult in
        (env3, ty { ft_mult = mult', ft_arg = ty1', ft_res = ty2' })

    go env (ForAllTy (Bndr tv vis) ty)
      = let (env1, subst', tv') = coreFlattenVarBndr subst env tv
            (env2, ty') = coreFlattenTy subst' env1 ty in
        (env2, ForAllTy (Bndr tv' vis) ty')

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

    go env (CastTy ty co)
      = let (env1, ty') = go env ty
            (env2, co') = coreFlattenCo subst env1 co in
        (env2, CastTy ty' co')

    go env (CoercionTy co)
      = let (env', co') = coreFlattenCo subst env co in
        (env', CoercionTy co')


-- when flattening, we don't care about the contents of coercions.
-- so, just return a fresh variable of the right (flattened) type
coreFlattenCo :: TvSubstEnv -> FlattenEnv
              -> Coercion -> (FlattenEnv, Coercion)
coreFlattenCo subst env co
  = (env2, mkCoVarCo covar)
  where
    (env1, kind') = coreFlattenTy subst env (coercionType co)
    covar         = mkFlattenFreshCoVar (fe_in_scope env1) kind'
    -- Add the covar to the FlattenEnv's in-scope set.
    -- See Note [Flattening type-family applications when matching instances], wrinkle 2A.
    env2          = updateInScopeSet env1 (flip extendInScopeSet covar)

coreFlattenVarBndr :: TvSubstEnv -> FlattenEnv
                   -> TyCoVar -> (FlattenEnv, TvSubstEnv, TyVar)
coreFlattenVarBndr subst env tv
  = (env2, subst', tv')
  where
    -- See Note [Flattening type-family applications when matching instances], wrinkle 2B.
    kind          = varType tv
    (env1, kind') = coreFlattenTy subst env kind
    tv'           = uniqAway (fe_in_scope env1) (setVarType tv kind')
    subst'        = extendVarEnv subst tv (mkTyVarTy tv')
    env2          = updateInScopeSet env1 (flip extendInScopeSet tv')

coreFlattenTyFamApp :: TvSubstEnv -> FlattenEnv
                    -> TyCon         -- type family tycon
                    -> [Type]        -- args, already flattened
                    -> (FlattenEnv, Type)
coreFlattenTyFamApp tv_subst env fam_tc fam_args
  = case lookupTypeMap type_map fam_ty of
      Just (tv, _, _) -> (env', mkAppTys (mkTyVarTy tv) leftover_args')
      Nothing ->
        let tyvar_name = mkFlattenFreshTyName fam_tc
            tv         = uniqAway in_scope $
                         mkTyVar tyvar_name (typeKind fam_ty)

            ty'   = mkAppTys (mkTyVarTy tv) leftover_args'
            env'' = env' { fe_type_map = extendTypeMap type_map fam_ty
                                                       (tv, fam_tc, sat_fam_args)
                         , fe_in_scope = extendInScopeSet in_scope tv }
        in (env'', ty')
  where
    arity = tyConArity fam_tc
    tcv_subst = Subst (fe_in_scope env) emptyIdSubstEnv tv_subst emptyVarEnv
    (sat_fam_args, leftover_args) = assert (arity <= length fam_args) $
                                    splitAt arity fam_args
    -- Apply the substitution before looking up an application in the
    -- environment. See Note [Flattening type-family applications when matching instances],
    -- wrinkle 1.
    -- NB: substTys short-cuts the common case when the substitution is empty.
    sat_fam_args' = substTys tcv_subst sat_fam_args
    (env', leftover_args') = coreFlattenTys tv_subst env leftover_args
    -- `fam_tc` may be over-applied to `fam_args` (see
    -- Note [Flattening type-family applications when matching instances]
    -- wrinkle 3), so we split it into the arguments needed to saturate it
    -- (sat_fam_args') and the rest (leftover_args')
    fam_ty = mkTyConApp fam_tc sat_fam_args'
    FlattenEnv { fe_type_map = type_map
               , fe_in_scope = in_scope } = env'

mkFlattenFreshTyName :: Uniquable a => a -> Name
mkFlattenFreshTyName unq
  = mkSysTvName (getUnique unq) (fsLit "flt")

mkFlattenFreshCoVar :: InScopeSet -> Kind -> CoVar
mkFlattenFreshCoVar in_scope kind
  = let uniq = unsafeGetFreshLocalUnique in_scope
        name = mkSystemVarName uniq (fsLit "flc")
    in mkCoVar name kind

