    -- (c) The University of Glasgow 2006

    {-# LANGUAGE CPP, DeriveDataTypeable #-}

    -- | Module for (a) type kinds and (b) type coercions,
    -- as used in System FC. See 'CoreSyn.Expr' for
    -- more on System FC and how coercions fit into it.
    --
    module Coercion (
            -- * Main data type
            Coercion(..), Var, CoVar,
            LeftOrRight(..), pickLR,
            Role(..), ltRole,

            -- ** Functions over coercions
            coVarKind, coVarRole,
            coercionType, coercionKind, coercionKinds, isReflCo,
            isReflCo_maybe, coercionRole, coercionKindRole,
            mkCoercionType,

            -- ** Constructing coercions
            mkReflCo, mkCoVarCo,
            mkAxInstCo, mkUnbranchedAxInstCo, mkAxInstLHS, mkAxInstRHS,
            mkUnbranchedAxInstRHS,
            mkPiCo, mkPiCos, mkCoCast,
            mkSymCo, mkTransCo, mkNthCo, mkNthCoRole, mkLRCo,
            mkInstCo, mkAppCo, mkAppCoFlexible, mkTyConAppCo, mkFunCo,
            mkForAllCo, mkUnsafeCo, mkUnivCo, mkSubCo, mkPhantomCo,
            mkNewTypeCo, downgradeRole,
            mkAxiomRuleCo,

            -- ** Decomposition
            instNewTyCon_maybe,

            NormaliseStepper, NormaliseStepResult(..), composeSteppers,
            modifyStepResultCo, unwrapNewTypeStepper,
            topNormaliseNewType_maybe, topNormaliseTypeX_maybe,

            decomposeCo, getCoVar_maybe,
            splitAppCo_maybe,
            splitForAllCo_maybe,
            nthRole, tyConRolesX,
            setNominalRole_maybe,

            -- ** Coercion variables
            mkCoVar, isCoVar, isCoVarType, coVarName, setCoVarName, setCoVarUnique,

            -- ** Free variables
            tyCoVarsOfCo, tyCoVarsOfCos, coVarsOfCo, coercionSize,

            -- ** Substitution
            CvSubstEnv, emptyCvSubstEnv,
            CvSubst(..), emptyCvSubst, Coercion.lookupTyVar, lookupCoVar,
            isEmptyCvSubst, zapCvSubstEnv, getCvInScope,
            substCo, substCos, substCoVar, substCoVars,
            substCoWithTy, substCoWithTys,
            cvTvSubst, tvCvSubst, mkCvSubst, zipOpenCvSubst,
            substTy, extendTvSubst,
            extendCvSubstAndInScope, extendTvSubstAndInScope,
            substTyVarBndr, substCoVarBndr,

            -- ** Lifting
            liftCoMatch, liftCoSubstTyVar, liftCoSubstWith,

            -- ** Comparison
            coreEqCoercion, coreEqCoercion2,

            -- ** Forcing evaluation of coercions
            seqCo,

            -- * Pretty-printing
            pprCo, pprParendCo,
            pprCoAxiom, pprCoAxBranch, pprCoAxBranchHdr,

            -- * Tidying
            tidyCo, tidyCos,

            -- * Other
            applyCo,
           ) where

    #include "HsVersions.h"

    import Unify    ( MatchEnv(..), matchList )
    import TypeRep
    import qualified Type
    import Type hiding( substTy, substTyVarBndr, extendTvSubst )
    import TyCon
    import CoAxiom
    import Var
    import VarEnv
    import VarSet
    import Binary
    import Maybes   ( orElse )
    import Name     ( Name, NamedThing(..), nameUnique, nameModule, getSrcSpan )
    import OccName  ( parenSymOcc )
    import Util
    import BasicTypes
    import Outputable
    import Unique
    import Pair
    import SrcLoc
    import PrelNames        ( funTyConKey, eqPrimTyConKey, eqReprPrimTyConKey )
    #if __GLASGOW_HASKELL__ < 709
    import Control.Applicative hiding ( empty )
    import Data.Traversable (traverse, sequenceA)
    #endif
    import FastString
    import ListSetOps

    import qualified Data.Data as Data hiding ( TyCon )
    import Control.Arrow ( first )

    {-
    ************************************************************************
    *                                                                      *
                Coercions
    *                                                                      *
    ************************************************************************
    -}

    -- | A 'Coercion' is concrete evidence of the equality/convertibility
    -- of two types.

    -- If you edit this type, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in coreSyn/CoreLint.lhs
    data Coercion
      -- Each constructor has a "role signature", indicating the way roles are
      -- propagated through coercions. P, N, and R stand for coercions of the
      -- given role. e stands for a coercion of a specific unknown role (think
      -- "role polymorphism"). "e" stands for an explicit role parameter
      -- indicating role e. _ stands for a parameter that is not a Role or
      -- Coercion.

      -- These ones mirror the shape of types
      = -- Refl :: "e" -> _ -> e
        Refl Role Type  -- See Note [Refl invariant]
              -- Invariant: applications of (Refl T) to a bunch of identity coercions
              --            always show up as Refl.
              -- For example  (Refl T) (Refl a) (Refl b) shows up as (Refl (T a b)).

              -- Applications of (Refl T) to some coercions, at least one of
              -- which is NOT the identity, show up as TyConAppCo.
              -- (They may not be fully saturated however.)
              -- ConAppCo coercions (like all coercions other than Refl)
              -- are NEVER the identity.

              -- Use (Refl Representational _), not (SubCo (Refl Nominal _))

      -- These ones simply lift the correspondingly-named
      -- Type constructors into Coercions

      -- TyConAppCo :: "e" -> _ -> ?? -> e
      -- See Note [TyConAppCo roles]
      | TyConAppCo Role TyCon [Coercion]    -- lift TyConApp
                   -- The TyCon is never a synonym;
                   -- we expand synonyms eagerly
                   -- But it can be a type function

      | AppCo Coercion Coercion        -- lift AppTy
              -- AppCo :: e -> N -> e

      -- See Note [Forall coercions]
      | ForAllCo TyVar Coercion       -- forall a. g
             -- :: _ -> e -> e

      -- These are special
      | CoVarCo CoVar      -- :: _ -> (N or R)
                           -- result role depends on the tycon of the variable's type

        -- AxiomInstCo :: e -> _ -> [N] -> e
      | AxiomInstCo (CoAxiom Branched) BranchIndex [Coercion]
         -- See also [CoAxiom index]
         -- The coercion arguments always *precisely* saturate
         -- arity of (that branch of) the CoAxiom.  If there are
         -- any left over, we use AppCo.  See
         -- See [Coercion axioms applied to coercions]

             -- see Note [UnivCo]
      | UnivCo FastString Role Type Type -- :: "e" -> _ -> _ -> e
                                   -- the FastString is just a note for provenance
      | SymCo Coercion             -- :: e -> e
      | TransCo Coercion Coercion  -- :: e -> e -> e

        -- The number of types and coercions should match exactly the expectations
        -- of the CoAxiomRule (i.e., the rule is fully saturated).
      | AxiomRuleCo CoAxiomRule [Type] [Coercion]

      -- These are destructors

      | NthCo  Int         Coercion     -- Zero-indexed; decomposes (T t0 ... tn)
        -- :: _ -> e -> ?? (inverse of TyConAppCo, see Note [TyConAppCo roles])
        -- See Note [NthCo and newtypes]

      | LRCo   LeftOrRight Coercion     -- Decomposes (t_left t_right)
        -- :: _ -> N -> N
      | InstCo Coercion Type
        -- :: e -> _ -> e

      | SubCo Coercion                  -- Turns a ~N into a ~R
        -- :: N -> R
      deriving (Data.Data, Data.Typeable)

    -- If you edit this type, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in coreSyn/CoreLint.lhs
    data LeftOrRight = CLeft | CRight
                     deriving( Eq, Data.Data, Data.Typeable )

    instance Binary LeftOrRight where
       put_ bh CLeft  = putByte bh 0
       put_ bh CRight = putByte bh 1

       get bh = do { h <- getByte bh
                   ; case h of
                       0 -> return CLeft
                       _ -> return CRight }

    pickLR :: LeftOrRight -> (a,a) -> a
    pickLR CLeft  (l,_) = l
    pickLR CRight (_,r) = r

    {-
    Note [Refl invariant]
    ~~~~~~~~~~~~~~~~~~~~~
    Coercions have the following invariant
         Refl is always lifted as far as possible.

    You might think that a consequencs is:
         Every identity coercions has Refl at the root

    But that's not quite true because of coercion variables.  Consider
         g         where g :: Int~Int
         Left h    where h :: Maybe Int ~ Maybe Int
    etc.  So the consequence is only true of coercions that
    have no coercion variables.

    Note [Coercion axioms applied to coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The reason coercion axioms can be applied to coercions and not just
    types is to allow for better optimization.  There are some cases where
    we need to be able to "push transitivity inside" an axiom in order to
    expose further opportunities for optimization.

    For example, suppose we have

      C a : t[a] ~ F a
      g   : b ~ c

    and we want to optimize

      sym (C b) ; t[g] ; C c

    which has the kind

      F b ~ F c

    (stopping through t[b] and t[c] along the way).

    We'd like to optimize this to just F g -- but how?  The key is
    that we need to allow axioms to be instantiated by *coercions*,
    not just by types.  Then we can (in certain cases) push
    transitivity inside the axiom instantiations, and then react
    opposite-polarity instantiations of the same axiom.  In this
    case, e.g., we match t[g] against the LHS of (C c)'s kind, to
    obtain the substitution  a |-> g  (note this operation is sort
    of the dual of lifting!) and hence end up with

      C g : t[b] ~ F c

    which indeed has the same kind as  t[g] ; C c.

    Now we have

      sym (C b) ; C g

    which can be optimized to F g.

    Note [CoAxiom index]
    ~~~~~~~~~~~~~~~~~~~~
    A CoAxiom has 1 or more branches. Each branch has contains a list
    of the free type variables in that branch, the LHS type patterns,
    and the RHS type for that branch. When we apply an axiom to a list
    of coercions, we must choose which branch of the axiom we wish to
    use, as the different branches may have different numbers of free
    type variables. (The number of type patterns is always the same
    among branches, but that doesn't quite concern us here.)

    The Int in the AxiomInstCo constructor is the 0-indexed number
    of the chosen branch.

    Note [Forall coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~
    Constructing coercions between forall-types can be a bit tricky.
    Currently, the situation is as follows:

      ForAllCo TyVar Coercion

    represents a coercion between polymorphic types, with the rule

               v : k       g : t1 ~ t2
      ----------------------------------------------
      ForAllCo v g : (all v:k . t1) ~ (all v:k . t2)

    Note that it's only necessary to coerce between polymorphic types
    where the type variables have identical kinds, because equality on
    kinds is trivial.

    Note [Predicate coercions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~
    Suppose we have
       g :: a~b
    How can we coerce between types
       ([c]~a) => [a] -> c
    and
       ([c]~b) => [b] -> c
    where the equality predicate *itself* differs?

    Answer: we simply treat (~) as an ordinary type constructor, so these
    types really look like

       ((~) [c] a) -> [a] -> c
       ((~) [c] b) -> [b] -> c

    So the coercion between the two is obviously

       ((~) [c] g) -> [g] -> c

    Another way to see this to say that we simply collapse predicates to
    their representation type (see Type.coreView and Type.predTypeRep).

    This collapse is done by mkPredCo; there is no PredCo constructor
    in Coercion.  This is important because we need Nth to work on
    predicates too:
        Nth 1 ((~) [c] g) = g
    See Simplify.simplCoercionF, which generates such selections.

    Note [Kind coercions]
    ~~~~~~~~~~~~~~~~~~~~~
    Suppose T :: * -> *, and g :: A ~ B
    Then the coercion
       TyConAppCo T [g]      T g : T A ~ T B

    Now suppose S :: forall k. k -> *, and g :: A ~ B
    Then the coercion
       TyConAppCo S [Refl *, g]   T <*> g : T * A ~ T * B

    Notice that the arguments to TyConAppCo are coercions, but the first
    represents a *kind* coercion. Now, we don't allow any non-trivial kind
    coercions, so it's an invariant that any such kind coercions are Refl.
    Lint checks this.

    However it's inconvenient to insist that these kind coercions are always
    *structurally* (Refl k), because the key function exprIsConApp_maybe
    pushes coercions into constructor arguments, so
           C k ty e |> g
    may turn into
           C (Nth 0 g) ....
    Now (Nth 0 g) will optimise to Refl, but perhaps not instantly.

    Note [Roles]
    ~~~~~~~~~~~~
    Roles are a solution to the GeneralizedNewtypeDeriving problem, articulated
    in Trac #1496. The full story is in docs/core-spec/core-spec.pdf. Also, see
    http://ghc.haskell.org/trac/ghc/wiki/RolesImplementation

    Here is one way to phrase the problem:

    Given:
    newtype Age = MkAge Int
    type family F x
    type instance F Age = Bool
    type instance F Int = Char

    This compiles down to:
    axAge :: Age ~ Int
    axF1 :: F Age ~ Bool
    axF2 :: F Int ~ Char

    Then, we can make:
    (sym (axF1) ; F axAge ; axF2) :: Bool ~ Char

    Yikes!

    The solution is _roles_, as articulated in "Generative Type Abstraction and
    Type-level Computation" (POPL 2010), available at
    http://www.seas.upenn.edu/~sweirich/papers/popl163af-weirich.pdf

    The specification for roles has evolved somewhat since that paper. For the
    current full details, see the documentation in docs/core-spec. Here are some
    highlights.

    We label every equality with a notion of type equivalence, of which there are
    three options: Nominal, Representational, and Phantom. A ground type is
    nominally equivalent only with itself. A newtype (which is considered a ground
    type in Haskell) is representationally equivalent to its representation.
    Anything is "phantomly" equivalent to anything else. We use "N", "R", and "P"
    to denote the equivalences.

    The axioms above would be:
    axAge :: Age ~R Int
    axF1 :: F Age ~N Bool
    axF2 :: F Age ~N Char

    Then, because transitivity applies only to coercions proving the same notion
    of equivalence, the above construction is impossible.

    However, there is still an escape hatch: we know that any two types that are
    nominally equivalent are representationally equivalent as well. This is what
    the form SubCo proves -- it "demotes" a nominal equivalence into a
    representational equivalence. So, it would seem the following is possible:

    sub (sym axF1) ; F axAge ; sub axF2 :: Bool ~R Char   -- WRONG

    What saves us here is that the arguments to a type function F, lifted into a
    coercion, *must* prove nominal equivalence. So, (F axAge) is ill-formed, and
    we are safe.

    Roles are attached to parameters to TyCons. When lifting a TyCon into a
    coercion (through TyConAppCo), we need to ensure that the arguments to the
    TyCon respect their roles. For example:

    data T a b = MkT a (F b)

    If we know that a1 ~R a2, then we know (T a1 b) ~R (T a2 b). But, if we know
    that b1 ~R b2, we know nothing about (T a b1) and (T a b2)! This is because
    the type function F branches on b's *name*, not representation. So, we say
    that 'a' has role Representational and 'b' has role Nominal. The third role,
    Phantom, is for parameters not used in the type's definition. Given the
    following definition

    data Q a = MkQ Int

    the Phantom role allows us to say that (Q Bool) ~R (Q Char), because we
    can construct the coercion Bool ~P Char (using UnivCo).

    See the paper cited above for more examples and information.

    Note [UnivCo]
    ~~~~~~~~~~~~~
    The UnivCo ("universal coercion") serves two rather separate functions:
     - the implementation for unsafeCoerce#
     - placeholder for phantom parameters in a TyConAppCo

    At Representational, it asserts that two (possibly unrelated)
    types have the same representation and can be casted to one another.
    This form is necessary for unsafeCoerce#.

    For optimisation purposes, it is convenient to allow UnivCo to appear
    at Nominal role. If we have

    data Foo a = MkFoo (F a)   -- F is a type family

    and we want an unsafe coercion from Foo Int to Foo Bool, then it would
    be nice to have (TyConAppCo Foo (UnivCo Nominal Int Bool)). So, we allow
    Nominal UnivCo's.

    At Phantom role, it is used as an argument to TyConAppCo in the place
    of a phantom parameter (a type parameter unused in the type definition).

    For example:

    data Q a = MkQ Int

    We want a coercion for (Q Bool) ~R (Q Char).

    (TyConAppCo Representational Q [UnivCo Phantom Bool Char]) does the trick.

    Note [TyConAppCo roles]
    ~~~~~~~~~~~~~~~~~~~~~~~
    The TyConAppCo constructor has a role parameter, indicating the role at
    which the coercion proves equality. The choice of this parameter affects
    the required roles of the arguments of the TyConAppCo. To help explain
    it, assume the following definition:

    newtype Age = MkAge Int

    Nominal: All arguments must have role Nominal. Why? So that Foo Age ~N Foo Int
    does *not* hold.

    Representational: All arguments must have the roles corresponding to the
    result of tyConRoles on the TyCon. This is the whole point of having
    roles on the TyCon to begin with. So, we can have Foo Age ~R Foo Int,
    if Foo's parameter has role R.

    If a Representational TyConAppCo is over-saturated (which is otherwise fine),
    the spill-over arguments must all be at Nominal. This corresponds to the
    behavior for AppCo.

    Phantom: All arguments must have role Phantom. This one isn't strictly
    necessary for soundness, but this choice removes ambiguity.

    The rules here also dictate what the parameters to mkTyConAppCo.

    Note [NthCo and newtypes]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    Suppose we have

      newtype N a = MkN Int
      type role N representational

    This yields axiom

      NTCo:N :: forall a. N a ~R Int

    We can then build

      co :: forall a b. N a ~R N b
      co = NTCo:N a ; sym (NTCo:N b)

    for any `a` and `b`. Because of the role annotation on N, if we use
    NthCo, we'll get out a representational coercion. That is:

      NthCo 0 co :: forall a b. a ~R b

    Yikes! Clearly, this is terrible. The solution is simple: forbid
    NthCo to be used on newtypes if the internal coercion is representational.

    This is not just some corner case discovered by a segfault somewhere;
    it was discovered in the proof of soundness of roles and described
    in the "Safe Coercions" paper (ICFP '14).

    ************************************************************************
    *                                                                      *
    \subsection{Coercion variables}
    *                                                                      *
    ************************************************************************
    -}

    coVarName :: CoVar -> Name
    coVarName = varName

    setCoVarUnique :: CoVar -> Unique -> CoVar
    setCoVarUnique = setVarUnique

    setCoVarName :: CoVar -> Name -> CoVar
    setCoVarName   = setVarName

    isCoVar :: Var -> Bool
    isCoVar v = isCoVarType (varType v)

    isCoVarType :: Type -> Bool
    isCoVarType ty      -- Tests for t1 ~# t2, the unboxed equality
      = case splitTyConApp_maybe ty of
          Just (tc,tys) -> (tc `hasKey` eqPrimTyConKey || tc `hasKey` eqReprPrimTyConKey)
                           && tys `lengthAtLeast` 2
          Nothing       -> False

    tyCoVarsOfCo :: Coercion -> VarSet
    -- Extracts type and coercion variables from a coercion
    tyCoVarsOfCo (Refl _ ty)           = tyVarsOfType ty
    tyCoVarsOfCo (TyConAppCo _ _ cos)  = tyCoVarsOfCos cos
    tyCoVarsOfCo (AppCo co1 co2)       = tyCoVarsOfCo co1 `unionVarSet` tyCoVarsOfCo co2
    tyCoVarsOfCo (ForAllCo tv co)      = tyCoVarsOfCo co `delVarSet` tv
    tyCoVarsOfCo (CoVarCo v)           = unitVarSet v
    tyCoVarsOfCo (AxiomInstCo _ _ cos) = tyCoVarsOfCos cos
    tyCoVarsOfCo (UnivCo _ _ ty1 ty2)  = tyVarsOfType ty1 `unionVarSet` tyVarsOfType ty2
    tyCoVarsOfCo (SymCo co)            = tyCoVarsOfCo co
    tyCoVarsOfCo (TransCo co1 co2)     = tyCoVarsOfCo co1 `unionVarSet` tyCoVarsOfCo co2
    tyCoVarsOfCo (NthCo _ co)          = tyCoVarsOfCo co
    tyCoVarsOfCo (LRCo _ co)           = tyCoVarsOfCo co
    tyCoVarsOfCo (InstCo co ty)        = tyCoVarsOfCo co `unionVarSet` tyVarsOfType ty
    tyCoVarsOfCo (SubCo co)            = tyCoVarsOfCo co
    tyCoVarsOfCo (AxiomRuleCo _ ts cs) = tyVarsOfTypes ts `unionVarSet` tyCoVarsOfCos cs

    tyCoVarsOfCos :: [Coercion] -> VarSet
    tyCoVarsOfCos = mapUnionVarSet tyCoVarsOfCo

    coVarsOfCo :: Coercion -> VarSet
    -- Extract *coerction* variables only.  Tiresome to repeat the code, but easy.
    coVarsOfCo (Refl _ _)            = emptyVarSet
    coVarsOfCo (TyConAppCo _ _ cos)  = coVarsOfCos cos
    coVarsOfCo (AppCo co1 co2)       = coVarsOfCo co1 `unionVarSet` coVarsOfCo co2
    coVarsOfCo (ForAllCo _ co)       = coVarsOfCo co
    coVarsOfCo (CoVarCo v)           = unitVarSet v
    coVarsOfCo (AxiomInstCo _ _ cos) = coVarsOfCos cos
    coVarsOfCo (UnivCo _ _ _ _)      = emptyVarSet
    coVarsOfCo (SymCo co)            = coVarsOfCo co
    coVarsOfCo (TransCo co1 co2)     = coVarsOfCo co1 `unionVarSet` coVarsOfCo co2
    coVarsOfCo (NthCo _ co)          = coVarsOfCo co
    coVarsOfCo (LRCo _ co)           = coVarsOfCo co
    coVarsOfCo (InstCo co _)         = coVarsOfCo co
    coVarsOfCo (SubCo co)            = coVarsOfCo co
    coVarsOfCo (AxiomRuleCo _ _ cos) = coVarsOfCos cos

    coVarsOfCos :: [Coercion] -> VarSet
    coVarsOfCos = mapUnionVarSet coVarsOfCo

    coercionSize :: Coercion -> Int
    coercionSize (Refl _ ty)           = typeSize ty
    coercionSize (TyConAppCo _ _ cos)  = 1 + sum (map coercionSize cos)
    coercionSize (AppCo co1 co2)       = coercionSize co1 + coercionSize co2
    coercionSize (ForAllCo _ co)       = 1 + coercionSize co
    coercionSize (CoVarCo _)           = 1
    coercionSize (AxiomInstCo _ _ cos) = 1 + sum (map coercionSize cos)
    coercionSize (UnivCo _ _ ty1 ty2)  = typeSize ty1 + typeSize ty2
    coercionSize (SymCo co)            = 1 + coercionSize co
    coercionSize (TransCo co1 co2)     = 1 + coercionSize co1 + coercionSize co2
    coercionSize (NthCo _ co)          = 1 + coercionSize co
    coercionSize (LRCo  _ co)          = 1 + coercionSize co
    coercionSize (InstCo co ty)        = 1 + coercionSize co + typeSize ty
    coercionSize (SubCo co)            = 1 + coercionSize co
    coercionSize (AxiomRuleCo _ tys cos) = 1 + sum (map typeSize tys)
                                             + sum (map coercionSize cos)

    {-
    ************************************************************************
    *                                                                      *
                                Tidying coercions
    *                                                                      *
    ************************************************************************
    -}

    tidyCo :: TidyEnv -> Coercion -> Coercion
    tidyCo env@(_, subst) co
      = go co
      where
        go (Refl r ty)            = Refl r (tidyType env ty)
        go (TyConAppCo r tc cos)  = let args = map go cos
                                    in args `seqList` TyConAppCo r tc args
        go (AppCo co1 co2)        = (AppCo $! go co1) $! go co2
        go (ForAllCo tv co)       = ForAllCo tvp $! (tidyCo envp co)
                                    where
                                      (envp, tvp) = tidyTyVarBndr env tv
        go (CoVarCo cv)           = case lookupVarEnv subst cv of
                                      Nothing  -> CoVarCo cv
                                      Just cv' -> CoVarCo cv'
        go (AxiomInstCo con ind cos) = let args = tidyCos env cos
                                       in args `seqList` AxiomInstCo con ind args
        go (UnivCo s r ty1 ty2)   = (UnivCo s r $! tidyType env ty1) $! tidyType env ty2
        go (SymCo co)             = SymCo $! go co
        go (TransCo co1 co2)      = (TransCo $! go co1) $! go co2
        go (NthCo d co)           = NthCo d $! go co
        go (LRCo lr co)           = LRCo lr $! go co
        go (InstCo co ty)         = (InstCo $! go co) $! tidyType env ty
        go (SubCo co)             = SubCo $! go co

        go (AxiomRuleCo ax tys cos) = let tys1 = map (tidyType env) tys
                                          cos1 = tidyCos env cos
                                      in tys1 `seqList` cos1 `seqList`
                                         AxiomRuleCo ax tys1 cos1

    tidyCos :: TidyEnv -> [Coercion] -> [Coercion]
    tidyCos env = map (tidyCo env)

    {-
    ************************************************************************
    *                                                                      *
                       Pretty-printing coercions
    *                                                                      *
    ************************************************************************

    @pprCo@ is the standard @Coercion@ printer; the overloaded @ppr@
    function is defined to use this.  @pprParendCo@ is the same, except it
    puts parens around the type, except for the atomic cases.
    @pprParendCo@ works just by setting the initial context precedence
    very high.
    -}

    instance Outputable Coercion where
      ppr = pprCo

    pprCo, pprParendCo :: Coercion -> SDoc
    pprCo       co = ppr_co TopPrec   co
    pprParendCo co = ppr_co TyConPrec co

    ppr_co :: TyPrec -> Coercion -> SDoc
    ppr_co _ (Refl r ty) = angleBrackets (ppr ty) <> ppr_role r

    ppr_co p co@(TyConAppCo _ tc [_,_])
      | tc `hasKey` funTyConKey = ppr_fun_co p co

    ppr_co _ (TyConAppCo r tc cos)  = pprTcApp TyConPrec ppr_co tc cos <> ppr_role r
    ppr_co p (AppCo co1 co2)        = maybeParen p TyConPrec $
                                      pprCo co1 <+> ppr_co TyConPrec co2
    ppr_co p co@(ForAllCo {})       = ppr_forall_co p co
    ppr_co _ (CoVarCo cv)           = parenSymOcc (getOccName cv) (ppr cv)
    ppr_co p (AxiomInstCo con index cos)
      = pprPrefixApp p (ppr (getName con) <> brackets (ppr index))
                       (map (ppr_co TyConPrec) cos)

    ppr_co p co@(TransCo {}) = maybeParen p FunPrec $
                               case trans_co_list co [] of
                                 [] -> panic "ppr_co"
                                 (co:cos) -> sep ( ppr_co FunPrec co
                                                 : [ char ';' <+> ppr_co FunPrec co | co <- cos])
    ppr_co p (InstCo co ty) = maybeParen p TyConPrec $
                              pprParendCo co <> ptext (sLit "@") <> pprType ty

    ppr_co p (UnivCo s r ty1 ty2) = pprPrefixApp p (ptext (sLit "UnivCo") <+> ftext s <+> ppr r)
                                               [pprParendType ty1, pprParendType ty2]
    ppr_co p (SymCo co)         = pprPrefixApp p (ptext (sLit "Sym")) [pprParendCo co]
    ppr_co p (NthCo n co)       = pprPrefixApp p (ptext (sLit "Nth:") <> int n) [pprParendCo co]
    ppr_co p (LRCo sel co)      = pprPrefixApp p (ppr sel) [pprParendCo co]
    ppr_co p (SubCo co)         = pprPrefixApp p (ptext (sLit "Sub")) [pprParendCo co]
    ppr_co p (AxiomRuleCo co ts cs) = maybeParen p TopPrec $
                                      ppr_axiom_rule_co co ts cs

    ppr_axiom_rule_co :: CoAxiomRule -> [Type] -> [Coercion] -> SDoc
    ppr_axiom_rule_co co ts ps = ppr (coaxrName co) <> ppTs ts $$ nest 2 (ppPs ps)
      where
      ppTs []   = Outputable.empty
      ppTs [t]  = ptext (sLit "@") <> ppr_type TopPrec t
      ppTs ts   = ptext (sLit "@") <>
                    parens (hsep $ punctuate comma $ map pprType ts)

      ppPs []   = Outputable.empty
      ppPs [p]  = pprParendCo p
      ppPs (p : ps) = ptext (sLit "(") <+> pprCo p $$
                      vcat [ ptext (sLit ",") <+> pprCo q | q <- ps ] $$
                      ptext (sLit ")")

    ppr_role :: Role -> SDoc
    ppr_role r = underscore <> pp_role
      where pp_role = case r of
                        Nominal          -> char 'N'
                        Representational -> char 'R'
                        Phantom          -> char 'P'

    trans_co_list :: Coercion -> [Coercion] -> [Coercion]
    trans_co_list (TransCo co1 co2) cos = trans_co_list co1 (trans_co_list co2 cos)
    trans_co_list co                cos = co : cos

    instance Outputable LeftOrRight where
      ppr CLeft    = ptext (sLit "Left")
      ppr CRight   = ptext (sLit "Right")

    ppr_fun_co :: TyPrec -> Coercion -> SDoc
    ppr_fun_co p co = pprArrowChain p (split co)
      where
        split :: Coercion -> [SDoc]
        split (TyConAppCo _ f [arg,res])
          | f `hasKey` funTyConKey
          = ppr_co FunPrec arg : split res
        split co = [ppr_co TopPrec co]

    ppr_forall_co :: TyPrec -> Coercion -> SDoc
    ppr_forall_co p ty
      = maybeParen p FunPrec $
        sep [pprForAll tvs, ppr_co TopPrec rho]
      where
        (tvs,  rho) = split1 [] ty
        split1 tvs (ForAllCo tv ty) = split1 (tv:tvs) ty
        split1 tvs ty               = (reverse tvs, ty)

    pprCoAxiom :: CoAxiom br -> SDoc
    pprCoAxiom ax@(CoAxiom { co_ax_tc = tc, co_ax_branches = branches })
      = hang (ptext (sLit "axiom") <+> ppr ax <+> dcolon)
           2 (vcat (map (pprCoAxBranch tc) $ fromBranchList branches))

    pprCoAxBranch :: TyCon -> CoAxBranch -> SDoc
    pprCoAxBranch fam_tc (CoAxBranch { cab_tvs = tvs
                                     , cab_lhs = lhs
                                     , cab_rhs = rhs })
      = hang (pprUserForAll tvs)
           2 (hang (pprTypeApp fam_tc lhs) 2 (equals <+> (ppr rhs)))

    pprCoAxBranchHdr :: CoAxiom br -> BranchIndex -> SDoc
    pprCoAxBranchHdr ax@(CoAxiom { co_ax_tc = fam_tc, co_ax_name = name }) index
      | CoAxBranch { cab_lhs = tys, cab_loc = loc } <- coAxiomNthBranch ax index
      = hang (pprTypeApp fam_tc tys)
           2 (ptext (sLit "-- Defined") <+> ppr_loc loc)
      where
            ppr_loc loc
              | isGoodSrcSpan loc
              = ptext (sLit "at") <+> ppr (srcSpanStart loc)

              | otherwise
              = ptext (sLit "in") <+>
                  quotes (ppr (nameModule name))

    {-
    ************************************************************************
    *                                                                      *
            Functions over Kinds
    *                                                                      *
    ************************************************************************
    -}

    -- | This breaks a 'Coercion' with type @T A B C ~ T D E F@ into
    -- a list of 'Coercion's of kinds @A ~ D@, @B ~ E@ and @E ~ F@. Hence:
    --
    -- > decomposeCo 3 c = [nth 0 c, nth 1 c, nth 2 c]
    decomposeCo :: Arity -> Coercion -> [Coercion]
    decomposeCo arity co
      = [mkNthCo n co | n <- [0..(arity-1)] ]
               -- Remember, Nth is zero-indexed

    -- | Attempts to obtain the type variable underlying a 'Coercion'
    getCoVar_maybe :: Coercion -> Maybe CoVar
    getCoVar_maybe (CoVarCo cv) = Just cv
    getCoVar_maybe _            = Nothing

    -- first result has role equal to input; second result is Nominal
    splitAppCo_maybe :: Coercion -> Maybe (Coercion, Coercion)
    -- ^ Attempt to take a coercion application apart.
    splitAppCo_maybe (AppCo co1 co2) = Just (co1, co2)
    splitAppCo_maybe (TyConAppCo r tc cos)
      | isDecomposableTyCon tc || cos `lengthExceeds` tyConArity tc
      , Just (cos', co') <- snocView cos
      , Just co'' <- setNominalRole_maybe co'
      = Just (mkTyConAppCo r tc cos', co'') -- Never create unsaturated type family apps!
           -- Use mkTyConAppCo to preserve the invariant
           --  that identity coercions are always represented by Refl
    splitAppCo_maybe (Refl r ty)
      | Just (ty1, ty2) <- splitAppTy_maybe ty
      = Just (Refl r ty1, Refl Nominal ty2)
    splitAppCo_maybe _ = Nothing

    splitForAllCo_maybe :: Coercion -> Maybe (TyVar, Coercion)
    splitForAllCo_maybe (ForAllCo tv co) = Just (tv, co)
    splitForAllCo_maybe _                = Nothing

    -------------------------------------------------------
    -- and some coercion kind stuff

    coVarKind :: CoVar -> (Type,Type)
    coVarKind cv
     | Just (tc, [_kind,ty1,ty2]) <- splitTyConApp_maybe (varType cv)
     = ASSERT(tc `hasKey` eqPrimTyConKey || tc `hasKey` eqReprPrimTyConKey)
       (ty1,ty2)
     | otherwise = panic "coVarKind, non coercion variable"

    coVarRole :: CoVar -> Role
    coVarRole cv
      | tc `hasKey` eqPrimTyConKey
      = Nominal
      | tc `hasKey` eqReprPrimTyConKey
      = Representational
      | otherwise
      = pprPanic "coVarRole: unknown tycon" (ppr cv)

      where
        tc = case tyConAppTyCon_maybe (varType cv) of
               Just tc0 -> tc0
               Nothing  -> pprPanic "coVarRole: not tyconapp" (ppr cv)

    -- | Makes a coercion type from two types: the types whose equality
    -- is proven by the relevant 'Coercion'
    mkCoercionType :: Role -> Type -> Type -> Type
    mkCoercionType Nominal          = mkPrimEqPred
    mkCoercionType Representational = mkReprPrimEqPred
    mkCoercionType Phantom          = panic "mkCoercionType"

    isReflCo :: Coercion -> Bool
    isReflCo (Refl {})         = True
    isReflCo _                 = False

    isReflCo_maybe :: Coercion -> Maybe Type
    isReflCo_maybe (Refl _ ty)       = Just ty
    isReflCo_maybe _                 = Nothing

    {-
    ************************************************************************
    *                                                                      *
                Building coercions
    *                                                                      *
    ************************************************************************

    Note [Role twiddling functions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    There are a plethora of functions for twiddling roles:

    mkSubCo: Requires a nominal input coercion and always produces a
    representational output. This is used when you (the programmer) are sure you
    know exactly that role you have and what you want.

    downgradeRole_maybe: This function takes both the input role and the output role
    as parameters. (The *output* role comes first!) It can only *downgrade* a
    role -- that is, change it from N to R or P, or from R to P. This one-way
    behavior is why there is the "_maybe". If an upgrade is requested, this
    function produces Nothing. This is used when you need to change the role of a
    coercion, but you're not sure (as you're writing the code) of which roles are
    involved.

    This function could have been written using coercionRole to ascertain the role
    of the input. But, that function is recursive, and the caller of downgradeRole_maybe
    often knows the input role. So, this is more efficient.

    downgradeRole: This is just like downgradeRole_maybe, but it panics if the conversion
    isn't a downgrade.

    setNominalRole_maybe: This is the only function that can *upgrade* a coercion. The result
    (if it exists) is always Nominal. The input can be at any role. It works on a
    "best effort" basis, as it should never be strictly necessary to upgrade a coercion
    during compilation. It is currently only used within GHC in splitAppCo_maybe. In order
    to be a proper inverse of mkAppCo, the second coercion that splitAppCo_maybe returns
    must be nominal. But, it's conceivable that splitAppCo_maybe is operating over a
    TyConAppCo that uses a representational coercion. Hence the need for setNominalRole_maybe.
    splitAppCo_maybe, in turn, is used only within coercion optimization -- thus, it is
    not absolutely critical that setNominalRole_maybe be complete.

    Note that setNominalRole_maybe will never upgrade a phantom UnivCo. Phantom
    UnivCos are perfectly type-safe, whereas representational and nominal ones are
    not. Indeed, `unsafeCoerce` is implemented via a representational UnivCo.
    (Nominal ones are no worse than representational ones, so this function *will*
    change a UnivCo Representational to a UnivCo Nominal.)

    Conal Elliott also came across a need for this function while working with the GHC
    API, as he was decomposing Core casts. The Core casts use representational coercions,
    as they must, but his use case required nominal coercions (he was building a GADT).
    So, that's why this function is exported from this module.

    One might ask: shouldn't downgradeRole_maybe just use setNominalRole_maybe as appropriate?
    I (Richard E.) have decided not to do this, because upgrading a role is bizarre and
    a caller should have to ask for this behavior explicitly.
    -}

    mkCoVarCo :: CoVar -> Coercion
    -- cv :: s ~# t
    mkCoVarCo cv
      | ty1 `eqType` ty2 = Refl (coVarRole cv) ty1
      | otherwise        = CoVarCo cv
      where
        (ty1, ty2) = ASSERT( isCoVar cv ) coVarKind cv

    mkReflCo :: Role -> Type -> Coercion
    mkReflCo = Refl

    mkAxInstCo :: Role -> CoAxiom br -> BranchIndex -> [Type] -> Coercion
    -- mkAxInstCo can legitimately be called over-staturated;
    -- i.e. with more type arguments than the coercion requires
    mkAxInstCo role ax index tys
      | arity == n_tys = downgradeRole role ax_role $ AxiomInstCo ax_br index rtys
      | otherwise      = ASSERT( arity < n_tys )
                         downgradeRole role ax_role $
                         foldl AppCo (AxiomInstCo ax_br index (take arity rtys))
                                     (drop arity rtys)
      where
        n_tys     = length tys
        ax_br     = toBranchedAxiom ax
        branch    = coAxiomNthBranch ax_br index
        arity     = length $ coAxBranchTyVars branch
        arg_roles = coAxBranchRoles branch
        rtys      = zipWith mkReflCo (arg_roles ++ repeat Nominal) tys
        ax_role   = coAxiomRole ax

    -- to be used only with unbranched axioms
    mkUnbranchedAxInstCo :: Role -> CoAxiom Unbranched -> [Type] -> Coercion
    mkUnbranchedAxInstCo role ax tys
      = mkAxInstCo role ax 0 tys

    mkAxInstLHS, mkAxInstRHS :: CoAxiom br -> BranchIndex -> [Type] -> Type
    -- Instantiate the axiom with specified types,
    -- returning the instantiated RHS
    -- A companion to mkAxInstCo:
    --    mkAxInstRhs ax index tys = snd (coercionKind (mkAxInstCo ax index tys))
    mkAxInstLHS ax index tys
      | CoAxBranch { cab_tvs = tvs, cab_lhs = lhs } <- coAxiomNthBranch ax index
      , (tys1, tys2) <- splitAtList tvs tys
      = ASSERT( tvs `equalLength` tys1 )
        mkTyConApp (coAxiomTyCon ax) (substTysWith tvs tys1 lhs ++ tys2)

    mkAxInstRHS ax index tys
      | CoAxBranch { cab_tvs = tvs, cab_rhs = rhs } <- coAxiomNthBranch ax index
      , (tys1, tys2) <- splitAtList tvs tys
      = ASSERT( tvs `equalLength` tys1 )
        mkAppTys (substTyWith tvs tys1 rhs) tys2

    mkUnbranchedAxInstRHS :: CoAxiom Unbranched -> [Type] -> Type
    mkUnbranchedAxInstRHS ax = mkAxInstRHS ax 0

    -- | Apply a 'Coercion' to another 'Coercion'.
    -- The second coercion must be Nominal, unless the first is Phantom.
    -- If the first is Phantom, then the second can be either Phantom or Nominal.
    mkAppCo :: Coercion -> Coercion -> Coercion
    mkAppCo co1 co2 = mkAppCoFlexible co1 Nominal co2
    -- Note, mkAppCo is careful to maintain invariants regarding
    -- where Refl constructors appear; see the comments in the definition
    -- of Coercion and the Note [Refl invariant] in types/TypeRep.lhs.

    -- | Apply a 'Coercion' to another 'Coercion'.
    -- The second 'Coercion's role is given, making this more flexible than
    -- 'mkAppCo'.
    mkAppCoFlexible :: Coercion -> Role -> Coercion -> Coercion
    mkAppCoFlexible (Refl r ty1) _ (Refl _ ty2)
      = Refl r (mkAppTy ty1 ty2)
    mkAppCoFlexible (Refl r ty1) r2 co2
      | Just (tc, tys) <- splitTyConApp_maybe ty1
        -- Expand type synonyms; a TyConAppCo can't have a type synonym (Trac #9102)
      = TyConAppCo r tc (zip_roles (tyConRolesX r tc) tys)
      where
        zip_roles (r1:_)  []        = [downgradeRole r1 r2 co2]
        zip_roles (r1:rs) (ty1:tys) = mkReflCo r1 ty1 : zip_roles rs tys
        zip_roles _       _         = panic "zip_roles" -- but the roles are infinite...
    mkAppCoFlexible (TyConAppCo r tc cos) r2 co
      = case r of
          Nominal          -> ASSERT( r2 == Nominal )
                              TyConAppCo Nominal tc (cos ++ [co])
          Representational -> TyConAppCo Representational tc (cos ++ [co'])
            where new_role = (tyConRolesX Representational tc) !! (length cos)
                  co'      = downgradeRole new_role r2 co
          Phantom          -> TyConAppCo Phantom tc (cos ++ [mkPhantomCo co])

    mkAppCoFlexible co1 _r2 co2 = ASSERT( _r2 == Nominal )
                                  AppCo co1 co2

    -- | Applies multiple 'Coercion's to another 'Coercion', from left to right.
    -- See also 'mkAppCo'.
    mkAppCos :: Coercion -> [Coercion] -> Coercion
    mkAppCos co1 cos = foldl mkAppCo co1 cos

    -- | Apply a type constructor to a list of coercions. It is the
    -- caller's responsibility to get the roles correct on argument coercions.
    mkTyConAppCo :: Role -> TyCon -> [Coercion] -> Coercion
    mkTyConAppCo r tc cos
                   -- Expand type synonyms
      | Just (tv_co_prs, rhs_ty, leftover_cos) <- tcExpandTyCon_maybe tc cos
      = mkAppCos (liftCoSubst r tv_co_prs rhs_ty) leftover_cos

      | Just tys <- traverse isReflCo_maybe cos
      = Refl r (mkTyConApp tc tys)  -- See Note [Refl invariant]

      | otherwise = TyConAppCo r tc cos

    -- | Make a function 'Coercion' between two other 'Coercion's
    mkFunCo :: Role -> Coercion -> Coercion -> Coercion
    mkFunCo r co1 co2 = mkTyConAppCo r funTyCon [co1, co2]

    -- | Make a 'Coercion' which binds a variable within an inner 'Coercion'
    mkForAllCo :: Var -> Coercion -> Coercion
    -- note that a TyVar should be used here, not a CoVar (nor a TcTyVar)
    mkForAllCo tv (Refl r ty)  = ASSERT( isTyVar tv ) Refl r (mkForAllTy tv ty)
    mkForAllCo tv  co          = ASSERT( isTyVar tv ) ForAllCo tv co

    -------------------------------

    -- | Create a symmetric version of the given 'Coercion' that asserts
    --   equality between the same types but in the other "direction", so
    --   a kind of @t1 ~ t2@ becomes the kind @t2 ~ t1@.
    mkSymCo :: Coercion -> Coercion

    -- Do a few simple optimizations, but don't bother pushing occurrences
    -- of symmetry to the leaves; the optimizer will take care of that.
    mkSymCo co@(Refl {})             = co
    mkSymCo    (UnivCo s r ty1 ty2)  = UnivCo s r ty2 ty1
    mkSymCo    (SymCo co)            = co
    mkSymCo co                       = SymCo co

    -- | Create a new 'Coercion' by composing the two given 'Coercion's transitively.
    mkTransCo :: Coercion -> Coercion -> Coercion
    mkTransCo (Refl {}) co = co
    mkTransCo co (Refl {}) = co
    mkTransCo co1 co2      = TransCo co1 co2

    -- the Role is the desired one. It is the caller's responsibility to make
    -- sure this request is reasonable
    mkNthCoRole :: Role -> Int -> Coercion -> Coercion
    mkNthCoRole role n co
      = downgradeRole role nth_role $ nth_co
      where
        nth_co = mkNthCo n co
        nth_role = coercionRole nth_co

    mkNthCo :: Int -> Coercion -> Coercion
    mkNthCo n (Refl r ty) = ASSERT( ok_tc_app ty n )
                            Refl r' (tyConAppArgN n ty)
      where tc = tyConAppTyCon ty
            r' = nthRole r tc n
    mkNthCo n co        = ASSERT( ok_tc_app _ty1 n && ok_tc_app _ty2 n )
                          NthCo n co
                        where
                          Pair _ty1 _ty2 = coercionKind co

    mkLRCo :: LeftOrRight -> Coercion -> Coercion
    mkLRCo lr (Refl eq ty) = Refl eq (pickLR lr (splitAppTy ty))
    mkLRCo lr co           = LRCo lr co

    ok_tc_app :: Type -> Int -> Bool
    ok_tc_app ty n = case splitTyConApp_maybe ty of
                       Just (_, tys) -> tys `lengthExceeds` n
                       Nothing       -> False

    -- | Instantiates a 'Coercion' with a 'Type' argument.
    mkInstCo :: Coercion -> Type -> Coercion
    mkInstCo co ty = InstCo co ty

    -- | Manufacture an unsafe coercion from thin air.
    --   Currently (May 14) this is used only to implement the
    --   @unsafeCoerce#@ primitive.  Optimise by pushing
    --   down through type constructors.
    mkUnsafeCo :: Type -> Type -> Coercion
    mkUnsafeCo = mkUnivCo (fsLit "mkUnsafeCo") Representational

    mkUnivCo :: FastString -> Role -> Type -> Type -> Coercion
    mkUnivCo prov role ty1 ty2
      | ty1 `eqType` ty2 = Refl role ty1
      | otherwise        = UnivCo prov role ty1 ty2

    mkAxiomRuleCo :: CoAxiomRule -> [Type] -> [Coercion] -> Coercion
    mkAxiomRuleCo = AxiomRuleCo

    -- input coercion is Nominal; see also Note [Role twiddling functions]
    mkSubCo :: Coercion -> Coercion
    mkSubCo (Refl Nominal ty) = Refl Representational ty
    mkSubCo (TyConAppCo Nominal tc cos)
      = TyConAppCo Representational tc (applyRoles tc cos)
    mkSubCo (UnivCo s Nominal ty1 ty2) = UnivCo s Representational ty1 ty2
    mkSubCo co = ASSERT2( coercionRole co == Nominal, ppr co <+> ppr (coercionRole co) )
                 SubCo co

    -- only *downgrades* a role. See Note [Role twiddling functions]
    downgradeRole_maybe :: Role   -- desired role
                        -> Role   -- current role
                        -> Coercion -> Maybe Coercion
    downgradeRole_maybe Representational Nominal co = Just (mkSubCo co)
    downgradeRole_maybe Nominal Representational _  = Nothing
    downgradeRole_maybe Phantom Phantom          co = Just co
    downgradeRole_maybe Phantom _                co = Just (mkPhantomCo co)
    downgradeRole_maybe _ Phantom                _  = Nothing
    downgradeRole_maybe _ _                      co = Just co

    -- panics if the requested conversion is not a downgrade.
    -- See also Note [Role twiddling functions]
    downgradeRole :: Role  -- desired role
                  -> Role  -- current role
                  -> Coercion -> Coercion
    downgradeRole r1 r2 co
      = case downgradeRole_maybe r1 r2 co of
          Just co' -> co'
          Nothing  -> pprPanic "downgradeRole" (ppr co)

    -- Converts a coercion to be nominal, if possible.
    -- See also Note [Role twiddling functions]
    setNominalRole_maybe :: Coercion -> Maybe Coercion
    setNominalRole_maybe co
      | Nominal <- coercionRole co = Just co
    setNominalRole_maybe (SubCo co)  = Just co
    setNominalRole_maybe (Refl _ ty) = Just $ Refl Nominal ty
    setNominalRole_maybe (TyConAppCo Representational tc coes)
      = do { cos' <- mapM setNominalRole_maybe coes
           ; return $ TyConAppCo Nominal tc cos' }
    setNominalRole_maybe (UnivCo s Representational ty1 ty2) = Just $ UnivCo s Nominal ty1 ty2
      -- We do *not* promote UnivCo Phantom, as that's unsafe.
      -- UnivCo Nominal is no more unsafe than UnivCo Representational
    setNominalRole_maybe (TransCo co1 co2)
      = TransCo <$> setNominalRole_maybe co1 <*> setNominalRole_maybe co2
    setNominalRole_maybe (AppCo co1 co2)
      = AppCo <$> setNominalRole_maybe co1 <*> pure co2
    setNominalRole_maybe (ForAllCo tv co)
      = ForAllCo tv <$> setNominalRole_maybe co
    setNominalRole_maybe (NthCo n co)
      = NthCo n <$> setNominalRole_maybe co
    setNominalRole_maybe (InstCo co ty)
      = InstCo <$> setNominalRole_maybe co <*> pure ty
    setNominalRole_maybe _ = Nothing

    -- takes any coercion and turns it into a Phantom coercion
    mkPhantomCo :: Coercion -> Coercion
    mkPhantomCo co
      | Just ty <- isReflCo_maybe co    = Refl Phantom ty
      | Pair ty1 ty2 <- coercionKind co = UnivCo (fsLit "mkPhantomCo") Phantom ty1 ty2
      -- don't optimise here... wait for OptCoercion

    -- All input coercions are assumed to be Nominal,
    -- or, if Role is Phantom, the Coercion can be Phantom, too.
    applyRole :: Role -> Coercion -> Coercion
    applyRole Nominal          = id
    applyRole Representational = mkSubCo
    applyRole Phantom          = mkPhantomCo

    -- Convert args to a TyConAppCo Nominal to the same TyConAppCo Representational
    applyRoles :: TyCon -> [Coercion] -> [Coercion]
    applyRoles tc cos
      = zipWith applyRole (tyConRolesX Representational tc) cos

    -- the Role parameter is the Role of the TyConAppCo
    -- defined here because this is intimiately concerned with the implementation
    -- of TyConAppCo
    tyConRolesX :: Role -> TyCon -> [Role]
    tyConRolesX Representational tc = tyConRoles tc ++ repeat Nominal
    tyConRolesX role             _  = repeat role

    nthRole :: Role -> TyCon -> Int -> Role
    nthRole Nominal _ _ = Nominal
    nthRole Phantom _ _ = Phantom
    nthRole Representational tc n
      = (tyConRolesX Representational tc) !! n

    ltRole :: Role -> Role -> Bool
    -- Is one role "less" than another?
    --     Nominal < Representational < Phantom
    ltRole Phantom          _       = False
    ltRole Representational Phantom = True
    ltRole Representational _       = False
    ltRole Nominal          Nominal = False
    ltRole Nominal          _       = True

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

    mkPiCos :: Role -> [Var] -> Coercion -> Coercion
    mkPiCos r vs co = foldr (mkPiCo r) co vs

    mkPiCo  :: Role -> Var -> Coercion -> Coercion
    mkPiCo r v co | isTyVar v = mkForAllCo v co
                  | otherwise = mkFunCo r (mkReflCo r (varType v)) co

    -- The first coercion *must* be Nominal.
    mkCoCast :: Coercion -> Coercion -> Coercion
    -- (mkCoCast (c :: s1 ~# t1) (g :: (s1 ~# t1) ~# (s2 ~# t2)
    mkCoCast c g
      = mkSymCo g1 `mkTransCo` c `mkTransCo` g2
      where
           -- g  :: (s1 ~# s2) ~# (t1 ~#  t2)
           -- g1 :: s1 ~# t1
           -- g2 :: s2 ~# t2
        [_reflk, g1, g2] = decomposeCo 3 g
                -- Remember, (~#) :: forall k. k -> k -> *
                -- so it takes *three* arguments, not two

    {-
    ************************************************************************
    *                                                                      *
                Newtypes
    *                                                                      *
    ************************************************************************
    -}

    -- | If @co :: T ts ~ rep_ty@ then:
    --
    -- > instNewTyCon_maybe T ts = Just (rep_ty, co)
    --
    -- Checks for a newtype, and for being saturated
    instNewTyCon_maybe :: TyCon -> [Type] -> Maybe (Type, Coercion)
    instNewTyCon_maybe tc tys
      | Just (tvs, ty, co_tc) <- unwrapNewTyConEtad_maybe tc  -- Check for newtype
      , tvs `leLength` tys                                    -- Check saturated enough
      = Just ( applyTysX tvs ty tys
             , mkUnbranchedAxInstCo Representational co_tc tys)
      | otherwise
      = Nothing

    {-
    ************************************************************************
    *                                                                      *
             Type normalisation
    *                                                                      *
    ************************************************************************
    -}

    -- | A function to check if we can reduce a type by one step. Used
    -- with 'topNormaliseTypeX_maybe'.
    type NormaliseStepper = RecTcChecker
                         -> TyCon     -- tc
                         -> [Type]    -- tys
                         -> NormaliseStepResult

    -- | The result of stepping in a normalisation function.
    -- See 'topNormaliseTypeX_maybe'.
    data NormaliseStepResult
      = NS_Done   -- ^ nothing more to do
      | NS_Abort  -- ^ utter failure. The outer function should fail too.
      | NS_Step RecTcChecker Type Coercion  -- ^ we stepped, yielding new bits;
                                            -- ^ co :: old type ~ new type

    modifyStepResultCo :: (Coercion -> Coercion)
                       -> NormaliseStepResult -> NormaliseStepResult
    modifyStepResultCo f (NS_Step rec_nts ty co) = NS_Step rec_nts ty (f co)
    modifyStepResultCo _ result                  = result

    -- | Try one stepper and then try the next, if the first doesn't make
    -- progress.
    composeSteppers :: NormaliseStepper -> NormaliseStepper
                    -> NormaliseStepper
    composeSteppers step1 step2 rec_nts tc tys
      = case step1 rec_nts tc tys of
          success@(NS_Step {}) -> success
          NS_Done              -> step2 rec_nts tc tys
          NS_Abort             -> NS_Abort

    -- | A 'NormaliseStepper' that unwraps newtypes, careful not to fall into
    -- a loop. If it would fall into a loop, it produces 'NS_Abort'.
    unwrapNewTypeStepper :: NormaliseStepper
    unwrapNewTypeStepper rec_nts tc tys
      | Just (ty', co) <- instNewTyCon_maybe tc tys
      = case checkRecTc rec_nts tc of
          Just rec_nts' -> NS_Step rec_nts' ty' co
          Nothing       -> NS_Abort

      | otherwise
      = NS_Done

    -- | A general function for normalising the top-level of a type. It continues
    -- to use the provided 'NormaliseStepper' until that function fails, and then
    -- this function returns. The roles of the coercions produced by the
    -- 'NormaliseStepper' must all be the same, which is the role returned from
    -- the call to 'topNormaliseTypeX_maybe'.
    topNormaliseTypeX_maybe :: NormaliseStepper -> Type -> Maybe (Coercion, Type)
    topNormaliseTypeX_maybe stepper
      = go initRecTc Nothing
      where
        go rec_nts mb_co1 ty
          | Just (tc, tys) <- splitTyConApp_maybe ty
          = case stepper rec_nts tc tys of
              NS_Step rec_nts' ty' co2
                -> go rec_nts' (mb_co1 `trans` co2) ty'

              NS_Done  -> all_done
              NS_Abort -> Nothing

          | otherwise
          = all_done
          where
            all_done | Just co <- mb_co1 = Just (co, ty)
                     | otherwise         = Nothing

        Nothing    `trans` co2 = Just co2
        (Just co1) `trans` co2 = Just (co1 `mkTransCo` co2)

    topNormaliseNewType_maybe :: Type -> Maybe (Coercion, Type)
    -- ^ Sometimes we want to look through a @newtype@ and get its associated coercion.
    -- This function strips off @newtype@ layers enough to reveal something that isn't
    -- a @newtype@, or responds False to ok_tc.  Specifically, here's the invariant:
    --
    -- > topNormaliseNewType_maybe ty = Just (co, ty')
    --
    -- then (a)  @co : ty0 ~ ty'@.
    --      (b)  ty' is not a newtype.
    --
    -- The function returns @Nothing@ for non-@newtypes@,
    -- or unsaturated applications
    --
    -- This function does *not* look through type families, because it has no access to
    -- the type family environment. If you do have that at hand, consider to use
    -- topNormaliseType_maybe, which should be a drop-in replacement for
    -- topNormaliseNewType_maybe
    --
    topNormaliseNewType_maybe ty
      = topNormaliseTypeX_maybe unwrapNewTypeStepper ty

    {-
    ************************************************************************
    *                                                                      *
                       Equality of coercions
    *                                                                      *
    ************************************************************************
    -}

    -- | Determines syntactic equality of coercions
    coreEqCoercion :: Coercion -> Coercion -> Bool
    coreEqCoercion co1 co2 = coreEqCoercion2 rn_env co1 co2
      where rn_env = mkRnEnv2 (mkInScopeSet (tyCoVarsOfCo co1 `unionVarSet` tyCoVarsOfCo co2))

    coreEqCoercion2 :: RnEnv2 -> Coercion -> Coercion -> Bool
    coreEqCoercion2 env (Refl eq1 ty1) (Refl eq2 ty2) = eq1 == eq2 && eqTypeX env ty1 ty2
    coreEqCoercion2 env (TyConAppCo eq1 tc1 cos1) (TyConAppCo eq2 tc2 cos2)
      = eq1 == eq2 && tc1 == tc2 && all2 (coreEqCoercion2 env) cos1 cos2

    coreEqCoercion2 env (AppCo co11 co12) (AppCo co21 co22)
      = coreEqCoercion2 env co11 co21 && coreEqCoercion2 env co12 co22

    coreEqCoercion2 env (ForAllCo v1 co1) (ForAllCo v2 co2)
      = coreEqCoercion2 (rnBndr2 env v1 v2) co1 co2

    coreEqCoercion2 env (CoVarCo cv1) (CoVarCo cv2)
      = rnOccL env cv1 == rnOccR env cv2

    coreEqCoercion2 env (AxiomInstCo con1 ind1 cos1) (AxiomInstCo con2 ind2 cos2)
      = con1 == con2
        && ind1 == ind2
        && all2 (coreEqCoercion2 env) cos1 cos2

    -- the provenance string is just a note, so don't use in comparisons
    coreEqCoercion2 env (UnivCo _ r1 ty11 ty12) (UnivCo _ r2 ty21 ty22)
      = r1 == r2 && eqTypeX env ty11 ty21 && eqTypeX env ty12 ty22

    coreEqCoercion2 env (SymCo co1) (SymCo co2)
      = coreEqCoercion2 env co1 co2

    coreEqCoercion2 env (TransCo co11 co12) (TransCo co21 co22)
      = coreEqCoercion2 env co11 co21 && coreEqCoercion2 env co12 co22

    coreEqCoercion2 env (NthCo d1 co1) (NthCo d2 co2)
      = d1 == d2 && coreEqCoercion2 env co1 co2
    coreEqCoercion2 env (LRCo d1 co1) (LRCo d2 co2)
      = d1 == d2 && coreEqCoercion2 env co1 co2

    coreEqCoercion2 env (InstCo co1 ty1) (InstCo co2 ty2)
      = coreEqCoercion2 env co1 co2 && eqTypeX env ty1 ty2

    coreEqCoercion2 env (SubCo co1) (SubCo co2)
      = coreEqCoercion2 env co1 co2

    coreEqCoercion2 env (AxiomRuleCo a1 ts1 cs1) (AxiomRuleCo a2 ts2 cs2)
      = a1 == a2 && all2 (eqTypeX env) ts1 ts2 && all2 (coreEqCoercion2 env) cs1 cs2

    coreEqCoercion2 _ _ _ = False

    {-
    ************************************************************************
    *                                                                      *
                       Substitution of coercions
    *                                                                      *
    ************************************************************************
    -}

    -- | A substitution of 'Coercion's for 'CoVar's (OR 'TyVar's, when
    --   doing a \"lifting\" substitution)
    type CvSubstEnv = VarEnv Coercion

    emptyCvSubstEnv :: CvSubstEnv
    emptyCvSubstEnv = emptyVarEnv

    data CvSubst
      = CvSubst InScopeSet  -- The in-scope type variables
                TvSubstEnv  -- Substitution of types
                CvSubstEnv  -- Substitution of coercions

    instance Outputable CvSubst where
      ppr (CvSubst ins tenv cenv)
        = brackets $ sep[ ptext (sLit "CvSubst"),
                          nest 2 (ptext (sLit "In scope:") <+> ppr ins),
                          nest 2 (ptext (sLit "Type env:") <+> ppr tenv),
                          nest 2 (ptext (sLit "Coercion env:") <+> ppr cenv) ]

    emptyCvSubst :: CvSubst
    emptyCvSubst = CvSubst emptyInScopeSet emptyVarEnv emptyVarEnv

    isEmptyCvSubst :: CvSubst -> Bool
    isEmptyCvSubst (CvSubst _ tenv cenv) = isEmptyVarEnv tenv && isEmptyVarEnv cenv

    getCvInScope :: CvSubst -> InScopeSet
    getCvInScope (CvSubst in_scope _ _) = in_scope

    zapCvSubstEnv :: CvSubst -> CvSubst
    zapCvSubstEnv (CvSubst in_scope _ _) = CvSubst in_scope emptyVarEnv emptyVarEnv

    cvTvSubst :: CvSubst -> TvSubst
    cvTvSubst (CvSubst in_scope tvs _) = TvSubst in_scope tvs

    tvCvSubst :: TvSubst -> CvSubst
    tvCvSubst (TvSubst in_scope tenv) = CvSubst in_scope tenv emptyCvSubstEnv

    extendTvSubst :: CvSubst -> TyVar -> Type -> CvSubst
    extendTvSubst (CvSubst in_scope tenv cenv) tv ty
      = CvSubst in_scope (extendVarEnv tenv tv ty) cenv

    extendTvSubstAndInScope :: CvSubst -> TyVar -> Type -> CvSubst
    extendTvSubstAndInScope (CvSubst in_scope tenv cenv) tv ty
      = CvSubst (in_scope `extendInScopeSetSet` tyVarsOfType ty)
                (extendVarEnv tenv tv ty)
                cenv

    extendCvSubstAndInScope :: CvSubst -> CoVar -> Coercion -> CvSubst
    -- Also extends the in-scope set
    extendCvSubstAndInScope (CvSubst in_scope tenv cenv) cv co
      = CvSubst (in_scope `extendInScopeSetSet` tyCoVarsOfCo co)
                tenv
                (extendVarEnv cenv cv co)

    substCoVarBndr :: CvSubst -> CoVar -> (CvSubst, CoVar)
    substCoVarBndr subst@(CvSubst in_scope tenv cenv) old_var
      = ASSERT( isCoVar old_var )
        (CvSubst (in_scope `extendInScopeSet` new_var) tenv new_cenv, new_var)
      where
        -- When we substitute (co :: t1 ~ t2) we may get the identity (co :: t ~ t)
        -- In that case, mkCoVarCo will return a ReflCoercion, and
        -- we want to substitute that (not new_var) for old_var
        new_co    = mkCoVarCo new_var
        no_change = new_var == old_var && not (isReflCo new_co)

        new_cenv | no_change = delVarEnv cenv old_var
                 | otherwise = extendVarEnv cenv old_var new_co

        new_var = uniqAway in_scope subst_old_var
        subst_old_var = mkCoVar (varName old_var) (substTy subst (varType old_var))
                      -- It's important to do the substitution for coercions,
                      -- because they can have free type variables

    substTyVarBndr :: CvSubst -> TyVar -> (CvSubst, TyVar)
    substTyVarBndr (CvSubst in_scope tenv cenv) old_var
      = case Type.substTyVarBndr (TvSubst in_scope tenv) old_var of
          (TvSubst in_scope' tenv', new_var) -> (CvSubst in_scope' tenv' cenv, new_var)

    mkCvSubst :: InScopeSet -> [(Var,Coercion)] -> CvSubst
    mkCvSubst in_scope prs = CvSubst in_scope Type.emptyTvSubstEnv (mkVarEnv prs)

    zipOpenCvSubst :: [Var] -> [Coercion] -> CvSubst
    zipOpenCvSubst vs cos
      | debugIsOn && (length vs /= length cos)
      = pprTrace "zipOpenCvSubst" (ppr vs $$ ppr cos) emptyCvSubst
      | otherwise
      = CvSubst (mkInScopeSet (tyCoVarsOfCos cos)) emptyTvSubstEnv (zipVarEnv vs cos)

    substCoWithTy :: InScopeSet -> TyVar -> Type -> Coercion -> Coercion
    substCoWithTy in_scope tv ty = substCoWithTys in_scope [tv] [ty]

    substCoWithTys :: InScopeSet -> [TyVar] -> [Type] -> Coercion -> Coercion
    substCoWithTys in_scope tvs tys co
      | debugIsOn && (length tvs /= length tys)
      = pprTrace "substCoWithTys" (ppr tvs $$ ppr tys) co
      | otherwise
      = ASSERT( length tvs == length tys )
        substCo (CvSubst in_scope (zipVarEnv tvs tys) emptyVarEnv) co

    -- | Substitute within a 'Coercion'
    substCo :: CvSubst -> Coercion -> Coercion
    substCo subst co | isEmptyCvSubst subst = co
                     | otherwise            = subst_co subst co

    -- | Substitute within several 'Coercion's
    substCos :: CvSubst -> [Coercion] -> [Coercion]
    substCos subst cos | isEmptyCvSubst subst = cos
                       | otherwise            = map (substCo subst) cos

    substTy :: CvSubst -> Type -> Type
    substTy subst = Type.substTy (cvTvSubst subst)

    subst_co :: CvSubst -> Coercion -> Coercion
    subst_co subst co
      = go co
      where
        go_ty :: Type -> Type
        go_ty = Coercion.substTy subst

        go :: Coercion -> Coercion
        go (Refl eq ty)          = Refl eq $! go_ty ty
        go (TyConAppCo eq tc cos)   = let args = map go cos
                                      in  args `seqList` TyConAppCo eq tc args
        go (AppCo co1 co2)       = mkAppCo (go co1) $! go co2
        go (ForAllCo tv co)      = case substTyVarBndr subst tv of
                                     (subst', tv') ->
                                       ForAllCo tv' $! subst_co subst' co
        go (CoVarCo cv)          = substCoVar subst cv
        go (AxiomInstCo con ind cos) = AxiomInstCo con ind $! map go cos
        go (UnivCo s r ty1 ty2)  = (UnivCo s r $! go_ty ty1) $! go_ty ty2
        go (SymCo co)            = mkSymCo (go co)
        go (TransCo co1 co2)     = mkTransCo (go co1) (go co2)
        go (NthCo d co)          = mkNthCo d (go co)
        go (LRCo lr co)          = mkLRCo lr (go co)
        go (InstCo co ty)        = mkInstCo (go co) $! go_ty ty
        go (SubCo co)            = mkSubCo (go co)
        go (AxiomRuleCo co ts cs)  = let ts1 = map go_ty ts
                                         cs1 = map go cs
                                     in ts1 `seqList` cs1 `seqList`
                                        AxiomRuleCo co ts1 cs1

    substCoVar :: CvSubst -> CoVar -> Coercion
    substCoVar (CvSubst in_scope _ cenv) cv
      | Just co  <- lookupVarEnv cenv cv      = co
      | Just cv1 <- lookupInScope in_scope cv = ASSERT( isCoVar cv1 ) CoVarCo cv1
      | otherwise = WARN( True, ptext (sLit "substCoVar not in scope") <+> ppr cv $$ ppr in_scope)
                    ASSERT( isCoVar cv ) CoVarCo cv

    substCoVars :: CvSubst -> [CoVar] -> [Coercion]
    substCoVars subst cvs = map (substCoVar subst) cvs

    lookupTyVar :: CvSubst -> TyVar  -> Maybe Type
    lookupTyVar (CvSubst _ tenv _) tv = lookupVarEnv tenv tv

    lookupCoVar :: CvSubst -> Var  -> Maybe Coercion
    lookupCoVar (CvSubst _ _ cenv) v = lookupVarEnv cenv v

    {-
    ************************************************************************
    *                                                                      *
                       "Lifting" substitution
               [(TyVar,Coercion)] -> Type -> Coercion
    *                                                                      *
    ************************************************************************

    Note [Lifting coercions over types: liftCoSubst]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The KPUSH rule deals with this situation
       data T a = MkK (a -> Maybe a)
       g :: T t1 ~ K t2
       x :: t1 -> Maybe t1

       case (K @t1 x) |> g of
         K (y:t2 -> Maybe t2) -> rhs

    We want to push the coercion inside the constructor application.
    So we do this

       g' :: t1~t2  =  Nth 0 g

       case K @t2 (x |> g' -> Maybe g') of
         K (y:t2 -> Maybe t2) -> rhs

    The crucial operation is that we
      * take the type of K's argument: a -> Maybe a
      * and substitute g' for a
    thus giving *coercion*.  This is what liftCoSubst does.

    Note [Substituting kinds in liftCoSubst]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We need to take care with kind polymorphism.  Suppose
      K :: forall k (a:k). (forall b:k. a -> b) -> T k a

    Now given  (K @kk1 @ty1 v) |> g) where
      g :: T kk1 ty1 ~ T kk2 ty2
    we want to compute
       (forall b:k a->b) [ Nth 0 g/k, Nth 1 g/a ]
    Notice that we MUST substitute for 'k'; this happens in
    liftCoSubstTyVarBndr.  But what should we substitute?
    We need to take b's kind 'k' and return a Kind, not a Coercion!

    Happily we can do this because we know that all kind coercions
    ((Nth 0 g) in this case) are Refl.  So we need a special purpose
       subst_kind: LiftCoSubst -> Kind -> Kind
    that expects a Refl coercion (or something equivalent to Refl)
    when it looks up a kind variable.
    -}

    -- ----------------------------------------------------
    -- See Note [Lifting coercions over types: liftCoSubst]
    -- ----------------------------------------------------

    data LiftCoSubst = LCS InScopeSet LiftCoEnv

    type LiftCoEnv = VarEnv Coercion
         -- Maps *type variables* to *coercions*
         -- That's the whole point of this function!

    liftCoSubstWith :: Role -> [TyVar] -> [Coercion] -> Type -> Coercion
    liftCoSubstWith r tvs cos ty
      = liftCoSubst r (zipEqual "liftCoSubstWith" tvs cos) ty

    liftCoSubst :: Role -> [(TyVar,Coercion)] -> Type -> Coercion
    liftCoSubst r prs ty
     | null prs  = Refl r ty
     | otherwise = ty_co_subst (LCS (mkInScopeSet (tyCoVarsOfCos (map snd prs)))
                                    (mkVarEnv prs)) r ty

    -- | The \"lifting\" operation which substitutes coercions for type
    --   variables in a type to produce a coercion.
    --
    --   For the inverse operation, see 'liftCoMatch'

    -- The Role parameter is the _desired_ role
    ty_co_subst :: LiftCoSubst -> Role -> Type -> Coercion
    ty_co_subst subst role ty
      = go role ty
      where
        go Phantom ty             = lift_phantom ty
        go role (TyVarTy tv)      = liftCoSubstTyVar subst role tv
                                    `orElse` Refl role (TyVarTy tv)
                                 -- A type variable from a non-cloned forall
                                 -- won't be in the substitution
        go role (AppTy ty1 ty2)   = mkAppCo (go role ty1) (go Nominal ty2)
        go role (TyConApp tc tys) = mkTyConAppCo role tc
                                               (zipWith go (tyConRolesX role tc) tys)
                               -- IA0_NOTE: Do we need to do anything
                               -- about kind instantiations? I don't think
                               -- so.  see Note [Kind coercions]
        go role (FunTy ty1 ty2)   = mkFunCo role (go role ty1) (go role ty2)
        go role (ForAllTy v ty)   = mkForAllCo v' $! (ty_co_subst subst' role ty)
                             where
                               (subst', v') = liftCoSubstTyVarBndr subst v
        go role ty@(LitTy {})     = ASSERT( role == Nominal )
                                    mkReflCo role ty

        lift_phantom ty = mkUnivCo (fsLit "lift_phantom")
                                   Phantom (liftCoSubstLeft  subst ty)
                                           (liftCoSubstRight subst ty)

    {-
    Note [liftCoSubstTyVar]
    ~~~~~~~~~~~~~~~~~~~~~~~
    This function can fail (i.e., return Nothing) for two separate reasons:
     1) The variable is not in the substutition
     2) The coercion found is of too low a role

    liftCoSubstTyVar is called from two places: in liftCoSubst (naturally), and
    also in matchAxiom in OptCoercion. From liftCoSubst, the so-called lifting
    lemma guarantees that the roles work out. If we fail for reason 2) in this
    case, we really should panic -- something is deeply wrong. But, in matchAxiom,
    failing for reason 2) is fine. matchAxiom is trying to find a set of coercions
    that match, but it may fail, and this is healthy behavior. Bottom line: if
    you find that liftCoSubst is doing weird things (like leaving out-of-scope
    variables lying around), disable coercion optimization (bypassing matchAxiom)
    and use downgradeRole instead of downgradeRole_maybe. The panic will then happen,
    and you may learn something useful.
    -}

    liftCoSubstTyVar :: LiftCoSubst -> Role -> TyVar -> Maybe Coercion
    liftCoSubstTyVar (LCS _ cenv) r tv
      = do { co <- lookupVarEnv cenv tv
           ; let co_role = coercionRole co   -- could theoretically take this as
                                             -- a parameter, but painful
           ; downgradeRole_maybe r co_role co } -- see Note [liftCoSubstTyVar]

    liftCoSubstTyVarBndr :: LiftCoSubst -> TyVar -> (LiftCoSubst, TyVar)
    liftCoSubstTyVarBndr subst@(LCS in_scope cenv) old_var
      = (LCS (in_scope `extendInScopeSet` new_var) new_cenv, new_var)
      where
        new_cenv | no_change = delVarEnv cenv old_var
                 | otherwise = extendVarEnv cenv old_var (Refl Nominal (TyVarTy new_var))

        no_change = no_kind_change && (new_var == old_var)

        new_var1 = uniqAway in_scope old_var

        old_ki = tyVarKind old_var
        no_kind_change = isEmptyVarSet (tyVarsOfType old_ki)
        new_var | no_kind_change = new_var1
                | otherwise      = setTyVarKind new_var1 (subst_kind subst old_ki)

    -- map every variable to the type on the *left* of its mapped coercion
    liftCoSubstLeft :: LiftCoSubst -> Type -> Type
    liftCoSubstLeft (LCS in_scope cenv) ty
      = Type.substTy (mkTvSubst in_scope (mapVarEnv (pFst . coercionKind) cenv)) ty

    -- same, but to the type on the right
    liftCoSubstRight :: LiftCoSubst -> Type -> Type
    liftCoSubstRight (LCS in_scope cenv) ty
      = Type.substTy (mkTvSubst in_scope (mapVarEnv (pSnd . coercionKind) cenv)) ty

    subst_kind :: LiftCoSubst -> Kind -> Kind
    -- See Note [Substituting kinds in liftCoSubst]
    subst_kind subst@(LCS _ cenv) kind
      = go kind
      where
        go (LitTy n)         = n `seq` LitTy n
        go (TyVarTy kv)      = subst_kv kv
        go (TyConApp tc tys) = let args = map go tys
                               in  args `seqList` TyConApp tc args

        go (FunTy arg res)   = (FunTy $! (go arg)) $! (go res)
        go (AppTy fun arg)   = mkAppTy (go fun) $! (go arg)
        go (ForAllTy tv ty)  = case liftCoSubstTyVarBndr subst tv of
                                  (subst', tv') ->
                                     ForAllTy tv' $! (subst_kind subst' ty)

        subst_kv kv
          | Just co <- lookupVarEnv cenv kv
          , let co_kind = coercionKind co
          = ASSERT2( pFst co_kind `eqKind` pSnd co_kind, ppr kv $$ ppr co )
            pFst co_kind
          | otherwise
          = TyVarTy kv

    -- | 'liftCoMatch' is sort of inverse to 'liftCoSubst'.  In particular, if
    --   @liftCoMatch vars ty co == Just s@, then @tyCoSubst s ty == co@.
    --   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.
    liftCoMatch :: TyVarSet -> Type -> Coercion -> Maybe LiftCoSubst
    liftCoMatch tmpls ty co
      = case ty_co_match menv emptyVarEnv ty co of
          Just cenv -> Just (LCS in_scope cenv)
          Nothing   -> Nothing
      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

    -- | 'ty_co_match' does all the actual work for 'liftCoMatch'.
    ty_co_match :: MatchEnv -> LiftCoEnv -> Type -> Coercion -> Maybe LiftCoEnv
    ty_co_match menv subst ty co
      | Just ty' <- coreView ty = ty_co_match menv subst ty' co

      -- Match a type variable against a non-refl coercion
    ty_co_match menv cenv (TyVarTy tv1) co
      | Just co1' <- lookupVarEnv cenv tv1'      -- tv1' is already bound to co1
      = if coreEqCoercion2 (nukeRnEnvL rn_env) co1' co
        then Just cenv
        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) (varSetElems (tyCoVarsOfCo co))
        then Nothing      -- occurs check failed
        else return (extendVarEnv cenv tv1' co)
            -- BAY: I don't think we need to do any kind matching here yet
            -- (compare 'match'), but we probably will when moving to SHE.

      | otherwise    -- tv1 is not a template ty var, so the only thing it
                     -- can match is a reflexivity coercion for itself.
                     -- But that case is dealt with already
      = Nothing

      where
        rn_env = me_env menv
        tv1' = rnOccL rn_env tv1

    ty_co_match menv subst (AppTy ty1 ty2) co
      | Just (co1, co2) <- splitAppCo_maybe co      -- c.f. Unify.match on AppTy
      = do { subst' <- ty_co_match menv subst ty1 co1
           ; ty_co_match menv subst' ty2 co2 }

    ty_co_match menv subst (TyConApp tc1 tys) (TyConAppCo _ tc2 cos)
      | tc1 == tc2 = ty_co_matches menv subst tys cos

    ty_co_match menv subst (FunTy ty1 ty2) (TyConAppCo _ tc cos)
      | tc == funTyCon = ty_co_matches menv subst [ty1,ty2] cos

    ty_co_match menv subst (ForAllTy tv1 ty) (ForAllCo tv2 co)
      = ty_co_match menv' subst ty co
      where
        menv' = menv { me_env = rnBndr2 (me_env menv) tv1 tv2 }

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

    ty_co_matches :: MatchEnv -> LiftCoEnv -> [Type] -> [Coercion] -> Maybe LiftCoEnv
    ty_co_matches menv = matchList (ty_co_match menv)

    pushRefl :: Coercion -> Maybe Coercion
    pushRefl (Refl Nominal (AppTy ty1 ty2))
      = Just (AppCo (Refl Nominal ty1) (Refl Nominal ty2))
    pushRefl (Refl r (FunTy ty1 ty2))
      = Just (TyConAppCo r funTyCon [Refl r ty1, Refl r ty2])
    pushRefl (Refl r (TyConApp tc tys))
      = Just (TyConAppCo r tc (zipWith mkReflCo (tyConRolesX r tc) tys))
    pushRefl (Refl r (ForAllTy tv ty)) = Just (ForAllCo tv (Refl r ty))
    pushRefl _                          = Nothing

    {-
    ************************************************************************
    *                                                                      *
                Sequencing on coercions
    *                                                                      *
    ************************************************************************
    -}

    seqCo :: Coercion -> ()
    seqCo (Refl eq ty)              = eq `seq` seqType ty
    seqCo (TyConAppCo eq tc cos)    = eq `seq` tc `seq` seqCos cos
    seqCo (AppCo co1 co2)           = seqCo co1 `seq` seqCo co2
    seqCo (ForAllCo tv co)          = tv `seq` seqCo co
    seqCo (CoVarCo cv)              = cv `seq` ()
    seqCo (AxiomInstCo con ind cos) = con `seq` ind `seq` seqCos cos
    seqCo (UnivCo s r ty1 ty2)      = s `seq` r `seq` seqType ty1 `seq` seqType ty2
    seqCo (SymCo co)                = seqCo co
    seqCo (TransCo co1 co2)         = seqCo co1 `seq` seqCo co2
    seqCo (NthCo _ co)              = seqCo co
    seqCo (LRCo _ co)               = seqCo co
    seqCo (InstCo co ty)            = seqCo co `seq` seqType ty
    seqCo (SubCo co)                = seqCo co
    seqCo (AxiomRuleCo _ ts cs)     = seqTypes ts `seq` seqCos cs

    seqCos :: [Coercion] -> ()
    seqCos []       = ()
    seqCos (co:cos) = seqCo co `seq` seqCos cos

    {-
    ************************************************************************
    *                                                                      *
                 The kind of a type, and of a coercion
    *                                                                      *
    ************************************************************************

    Note [Computing a coercion kind and role]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    To compute a coercion's kind is straightforward: see coercionKind.
    But to compute a coercion's role, in the case for NthCo we need
    its kind as well.  So if we have two separate functions (one for kinds
    and one for roles) we can get exponentially bad behaviour, since each
    NthCo node makes a separate call to coercionKind, which traverses the
    sub-tree again.  This was part of the problem in Trac #9233.

    Solution: compute both together; hence coercionKindRole.  We keep a
    separate coercionKind function because it's a bit more efficient if
    the kind is all you want.
    -}

    coercionType :: Coercion -> Type
    coercionType co = case coercionKindRole co of
                        (Pair ty1 ty2, r) -> mkCoercionType r ty1 ty2

    ------------------
    -- | If it is the case that
    --
    -- > c :: (t1 ~ t2)
    --
    -- i.e. the kind of @c@ relates @t1@ and @t2@, then @coercionKind c = Pair t1 t2@.

    coercionKind :: Coercion -> Pair Type
    coercionKind co = go co
      where
        go (Refl _ ty)           = Pair ty ty
        go (TyConAppCo _ tc cos) = mkTyConApp tc <$> (sequenceA $ map go cos)
        go (AppCo co1 co2)       = mkAppTy <$> go co1 <*> go co2
        go (ForAllCo tv co)      = mkForAllTy tv <$> go co
        go (CoVarCo cv)          = toPair $ coVarKind cv
        go (AxiomInstCo ax ind cos)
          | CoAxBranch { cab_tvs = tvs, cab_lhs = lhs, cab_rhs = rhs } <- coAxiomNthBranch ax ind
          , Pair tys1 tys2 <- sequenceA (map go cos)
          = ASSERT( cos `equalLength` tvs )  -- Invariant of AxiomInstCo: cos should
                                             -- exactly saturate the axiom branch
            Pair (substTyWith tvs tys1 (mkTyConApp (coAxiomTyCon ax) lhs))
                 (substTyWith tvs tys2 rhs)
        go (UnivCo _ _ ty1 ty2)  = Pair ty1 ty2
        go (SymCo co)            = swap $ go co
        go (TransCo co1 co2)     = Pair (pFst $ go co1) (pSnd $ go co2)
        go (NthCo d co)          = tyConAppArgN d <$> go co
        go (LRCo lr co)          = (pickLR lr . splitAppTy) <$> go co
        go (InstCo aco ty)       = go_app aco [ty]
        go (SubCo co)            = go co
        go (AxiomRuleCo ax tys cos) =
          case coaxrProves ax tys (map go cos) of
            Just res -> res
            Nothing  -> panic "coercionKind: Malformed coercion"

        go_app :: Coercion -> [Type] -> Pair Type
        -- Collect up all the arguments and apply all at once
        -- See Note [Nested InstCos]
        go_app (InstCo co ty) tys = go_app co (ty:tys)
        go_app co             tys = (`applyTys` tys) <$> go co

    -- | Apply 'coercionKind' to multiple 'Coercion's
    coercionKinds :: [Coercion] -> Pair [Type]
    coercionKinds tys = sequenceA $ map coercionKind tys

    -- | Get a coercion's kind and role.
    -- Why both at once?  See Note [Computing a coercion kind and role]
    coercionKindRole :: Coercion -> (Pair Type, Role)
    coercionKindRole = go
      where
        go (Refl r ty) = (Pair ty ty, r)
        go (TyConAppCo r tc cos)
          = (mkTyConApp tc <$> (sequenceA $ map coercionKind cos), r)
        go (AppCo co1 co2)
          = let (tys1, r1) = go co1 in
            (mkAppTy <$> tys1 <*> coercionKind co2, r1)
        go (ForAllCo tv co)
          = let (tys, r) = go co in
            (mkForAllTy tv <$> tys, r)
        go (CoVarCo cv) = (toPair $ coVarKind cv, coVarRole cv)
        go co@(AxiomInstCo ax _ _) = (coercionKind co, coAxiomRole ax)
        go (UnivCo _ r ty1 ty2) = (Pair ty1 ty2, r)
        go (SymCo co) = first swap $ go co
        go (TransCo co1 co2)
          = let (tys1, r) = go co1 in
            (Pair (pFst tys1) (pSnd $ coercionKind co2), r)
        go (NthCo d co)
          = let (Pair t1 t2, r) = go co
                (tc1,  args1) = splitTyConApp t1
                (_tc2, args2) = splitTyConApp t2
            in
            ASSERT( tc1 == _tc2 )
            ((`getNth` d) <$> Pair args1 args2, nthRole r tc1 d)
        go co@(LRCo {}) = (coercionKind co, Nominal)
        go (InstCo co ty) = go_app co [ty]
        go (SubCo co) = (coercionKind co, Representational)
        go co@(AxiomRuleCo ax _ _) = (coercionKind co, coaxrRole ax)

        go_app :: Coercion -> [Type] -> (Pair Type, Role)
        -- Collect up all the arguments and apply all at once
        -- See Note [Nested InstCos]
        go_app (InstCo co ty) tys = go_app co (ty:tys)
        go_app co             tys
          = let (pair, r) = go co in
            ((`applyTys` tys) <$> pair, r)

    -- | Retrieve the role from a coercion.
    coercionRole :: Coercion -> Role
    coercionRole = snd . coercionKindRole
      -- There's not a better way to do this, because NthCo needs the *kind*
      -- and role of its argument. Luckily, laziness should generally avoid
      -- the need for computing kinds in other cases.

    {-
    Note [Nested InstCos]
    ~~~~~~~~~~~~~~~~~~~~~
    In Trac #5631 we found that 70% of the entire compilation time was
    being spent in coercionKind!  The reason was that we had
       (g @ ty1 @ ty2 .. @ ty100)    -- The "@s" are InstCos
    where
       g :: forall a1 a2 .. a100. phi
    If we deal with the InstCos one at a time, we'll do this:
       1.  Find the kind of (g @ ty1 .. @ ty99) : forall a100. phi'
       2.  Substitute phi'[ ty100/a100 ], a single tyvar->type subst
    But this is a *quadratic* algorithm, and the blew up Trac #5631.
    So it's very important to do the substitution simultaneously.

    cf Type.applyTys (which in fact we call here)
    -}

    applyCo :: Type -> Coercion -> Type
    -- Gives the type of (e co) where e :: (a~b) => ty
    applyCo ty co | Just ty' <- coreView ty = applyCo ty' co
    applyCo (FunTy _ ty) _ = ty
    applyCo _            _ = panic "applyCo"

    {-
    Note [Kind coercions]
    ~~~~~~~~~~~~~~~~~~~~~
    Kind coercions are only of the form: Refl kind. They are only used to
    instantiate kind polymorphic type constructors in TyConAppCo. Remember
    that kind instantiation only happens with TyConApp, not AppTy.
    -}