%
% (c) The University of Glasgow 2006
%

\begin{code}
-- | 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.
--
-- Coercions are represented as types, and their kinds tell what types the 
-- coercion works on. The coercion kind constructor is a special TyCon that 
-- must always be saturated, like so:
--
-- > typeKind (symCoercion type) :: TyConApp CoTyCon{...} [type, type]
module Coercion (
        -- * Main data type
        Coercion, Kind,
        typeKind,

        -- ** Deconstructing Kinds 
        kindFunResult, kindAppResult, synTyConResKind,
        splitKindFunTys, splitKindFunTysN, splitKindFunTy_maybe,

        -- ** Predicates on Kinds
        isLiftedTypeKind, isUnliftedTypeKind, isOpenTypeKind,
        isUbxTupleKind, isArgTypeKind, isKind, isTySuperKind, 
        isCoSuperKind, isSuperKind, isCoercionKind, 
	mkArrowKind, mkArrowKinds,

        isSubArgTypeKind, isSubOpenTypeKind, isSubKind, defaultKind, eqKind,
        isSubKindCon,

        mkCoKind, mkCoPredTy, coVarKind, coVarKind_maybe,
        coercionKind, coercionKinds, isIdentityCoercion,

	-- ** Equality predicates
	isEqPred, mkEqPred, getEqPredTys, isEqPredTy,  

	-- ** Coercion transformations
	mkCoercion,
        mkSymCoercion, mkTransCoercion,
        mkLeftCoercion, mkRightCoercion, 
	mkInstCoercion, mkAppCoercion, mkTyConCoercion, mkFunCoercion,
        mkForAllCoercion, mkInstsCoercion, mkUnsafeCoercion,
        mkNewTypeCoercion, mkFamInstCoercion, mkAppsCoercion,
        mkCsel1Coercion, mkCsel2Coercion, mkCselRCoercion, 

	mkClassPPredCo, mkIParamPredCo, mkEqPredCo, 
        mkCoVarCoercion, mkCoPredCo, 


        unsafeCoercionTyCon, symCoercionTyCon,
        transCoercionTyCon, leftCoercionTyCon, 
        rightCoercionTyCon, instCoercionTyCon, -- needed by TysWiredIn
        csel1CoercionTyCon, csel2CoercionTyCon, cselRCoercionTyCon, 

        -- ** Decomposition
        decompLR_maybe, decompCsel_maybe, decompInst_maybe,
        splitCoPredTy_maybe,
        splitNewTypeRepCo_maybe, instNewTyCon_maybe, decomposeCo,

        -- ** Comparison
        coreEqCoercion, coreEqCoercion2,

	-- * CoercionI
	CoercionI(..),
	isIdentityCoI,
	mkSymCoI, mkTransCoI, 
	mkTyConAppCoI, mkAppTyCoI, mkFunTyCoI,
	mkForAllTyCoI,
	fromCoI, 
	mkClassPPredCoI, mkIParamPredCoI, mkEqPredCoI, mkCoPredCoI 

       ) where 

#include "HsVersions.h"

import TypeRep
import Type
import TyCon
import Class
import Var
import VarEnv
import VarSet
import Name
import PrelNames
import Util
import BasicTypes
import Outputable
import FastString
\end{code}

%************************************************************************
%*									*
	Functions over Kinds		
%*									*
%************************************************************************

\begin{code}
-- | Essentially 'funResultTy' on kinds
kindFunResult :: Kind -> Kind
kindFunResult k = funResultTy k

kindAppResult :: Kind -> [arg] -> Kind
kindAppResult k []     = k
kindAppResult k (_:as) = kindAppResult (kindFunResult k) as

-- | Essentially 'splitFunTys' on kinds
splitKindFunTys :: Kind -> ([Kind],Kind)
splitKindFunTys k = splitFunTys k

splitKindFunTy_maybe :: Kind -> Maybe (Kind,Kind)
splitKindFunTy_maybe = splitFunTy_maybe

-- | Essentially 'splitFunTysN' on kinds
splitKindFunTysN :: Int -> Kind -> ([Kind],Kind)
splitKindFunTysN k = splitFunTysN k

-- | Find the result 'Kind' of a type synonym, 
-- after applying it to its 'arity' number of type variables
-- Actually this function works fine on data types too, 
-- but they'd always return '*', so we never need to ask
synTyConResKind :: TyCon -> Kind
synTyConResKind tycon = kindAppResult (tyConKind tycon) (tyConTyVars tycon)

-- | See "Type#kind_subtyping" for details of the distinction between these 'Kind's
isUbxTupleKind, isOpenTypeKind, isArgTypeKind, isUnliftedTypeKind :: Kind -> Bool
isOpenTypeKindCon, isUbxTupleKindCon, isArgTypeKindCon,
        isUnliftedTypeKindCon, isSubArgTypeKindCon      :: TyCon -> Bool

isOpenTypeKindCon tc    = tyConUnique tc == openTypeKindTyConKey

isOpenTypeKind (TyConApp tc _) = isOpenTypeKindCon tc
isOpenTypeKind _               = False

isUbxTupleKindCon tc = tyConUnique tc == ubxTupleKindTyConKey

isUbxTupleKind (TyConApp tc _) = isUbxTupleKindCon tc
isUbxTupleKind _               = False

isArgTypeKindCon tc = tyConUnique tc == argTypeKindTyConKey

isArgTypeKind (TyConApp tc _) = isArgTypeKindCon tc
isArgTypeKind _               = False

isUnliftedTypeKindCon tc = tyConUnique tc == unliftedTypeKindTyConKey

isUnliftedTypeKind (TyConApp tc _) = isUnliftedTypeKindCon tc
isUnliftedTypeKind _               = False

isSubOpenTypeKind :: Kind -> Bool
-- ^ True of any sub-kind of OpenTypeKind (i.e. anything except arrow)
isSubOpenTypeKind (FunTy k1 k2)    = ASSERT2 ( isKind k1, text "isSubOpenTypeKind" <+> ppr k1 <+> text "::" <+> ppr (typeKind k1) ) 
                                     ASSERT2 ( isKind k2, text "isSubOpenTypeKind" <+> ppr k2 <+> text "::" <+> ppr (typeKind k2) ) 
                                     False
isSubOpenTypeKind (TyConApp kc []) = ASSERT( isKind (TyConApp kc []) ) True
isSubOpenTypeKind other            = ASSERT( isKind other ) False
         -- This is a conservative answer
         -- It matters in the call to isSubKind in
	 -- checkExpectedKind.

isSubArgTypeKindCon kc
  | isUnliftedTypeKindCon kc = True
  | isLiftedTypeKindCon kc   = True
  | isArgTypeKindCon kc      = True
  | otherwise                = False

isSubArgTypeKind :: Kind -> Bool
-- ^ True of any sub-kind of ArgTypeKind 
isSubArgTypeKind (TyConApp kc []) = isSubArgTypeKindCon kc
isSubArgTypeKind _                = False

-- | Is this a super-kind (i.e. a type-of-kinds)?
isSuperKind :: Type -> Bool
isSuperKind (TyConApp (skc) []) = isSuperKindTyCon skc
isSuperKind _                   = False

-- | Is this a kind (i.e. a type-of-types)?
isKind :: Kind -> Bool
isKind k = isSuperKind (typeKind k)

isSubKind :: Kind -> Kind -> Bool
-- ^ @k1 \`isSubKind\` k2@ checks that @k1@ <: @k2@
isSubKind (TyConApp kc1 []) (TyConApp kc2 []) = kc1 `isSubKindCon` kc2
isSubKind (FunTy a1 r1) (FunTy a2 r2)	      = (a2 `isSubKind` a1) && (r1 `isSubKind` r2)
isSubKind (PredTy (EqPred ty1 ty2)) (PredTy (EqPred ty1' ty2')) 
  = ty1 `tcEqType` ty1' && ty2 `tcEqType` ty2'
isSubKind _             _                     = False

eqKind :: Kind -> Kind -> Bool
eqKind = tcEqType

isSubKindCon :: TyCon -> TyCon -> Bool
-- ^ @kc1 \`isSubKindCon\` kc2@ checks that @kc1@ <: @kc2@
isSubKindCon kc1 kc2
  | isLiftedTypeKindCon kc1   && isLiftedTypeKindCon kc2   = True
  | isUnliftedTypeKindCon kc1 && isUnliftedTypeKindCon kc2 = True
  | isUbxTupleKindCon kc1     && isUbxTupleKindCon kc2     = True
  | isOpenTypeKindCon kc2                                  = True 
                           -- we already know kc1 is not a fun, its a TyCon
  | isArgTypeKindCon kc2      && isSubArgTypeKindCon kc1   = True
  | otherwise                                              = False

defaultKind :: Kind -> Kind
-- ^ Used when generalising: default kind ? and ?? to *. See "Type#kind_subtyping" for more
-- information on what that means

-- When we generalise, we make generic type variables whose kind is
-- simple (* or *->* etc).  So generic type variables (other than
-- built-in constants like 'error') always have simple kinds.  This is important;
-- consider
--	f x = True
-- We want f to get type
--	f :: forall (a::*). a -> Bool
-- Not 
--	f :: forall (a::??). a -> Bool
-- because that would allow a call like (f 3#) as well as (f True),
--and the calling conventions differ.  This defaulting is done in TcMType.zonkTcTyVarBndr.
defaultKind k 
  | isSubOpenTypeKind k = liftedTypeKind
  | isSubArgTypeKind k  = liftedTypeKind
  | otherwise        = k
\end{code}

%************************************************************************
%*									*
            Coercions
%*									*
%************************************************************************


\begin{code}
-- | A 'Coercion' represents a 'Type' something should be coerced to.
type Coercion     = Type

-- | A 'CoercionKind' is always of form @ty1 ~ ty2@ and indicates the
-- types that a 'Coercion' will work on.
type CoercionKind = Kind

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

-- | This breaks a 'Coercion' with 'CoercionKind' @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 = [right (left (left c)), right (left c), right c]
decomposeCo :: Arity -> Coercion -> [Coercion]
decomposeCo n co
  = go n co []
  where
    go 0 _  cos = cos
    go n co cos = go (n-1) (mkLeftCoercion co)
			   (mkRightCoercion co : cos)


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

coVarKind :: CoVar -> (Type,Type) 
-- c :: t1 ~ t2
coVarKind cv = case coVarKind_maybe cv of
                 Just ts -> ts
                 Nothing -> pprPanic "coVarKind" (ppr cv $$ ppr (tyVarKind cv))

coVarKind_maybe :: CoVar -> Maybe (Type,Type) 
coVarKind_maybe cv = splitCoKind_maybe (tyVarKind cv)

-- | Take a 'CoercionKind' apart into the two types it relates: see also 'mkCoKind'.
-- Panics if the argument is not a valid 'CoercionKind'
splitCoKind_maybe :: Kind -> Maybe (Type, Type)
splitCoKind_maybe co | Just co' <- kindView co = splitCoKind_maybe co'
splitCoKind_maybe (PredTy (EqPred ty1 ty2))    = Just (ty1, ty2)
splitCoKind_maybe _                            = Nothing

-- | Makes a 'CoercionKind' from two types: the types whose equality 
-- is proven by the relevant 'Coercion'
mkCoKind :: Type -> Type -> CoercionKind
mkCoKind ty1 ty2 = PredTy (EqPred ty1 ty2)

-- | (mkCoPredTy s t r) produces the type:   (s~t) => r
mkCoPredTy :: Type -> Type -> Type -> Type
mkCoPredTy s t r = ASSERT( not (co_var `elemVarSet` tyVarsOfType r) )
                   ForAllTy co_var r
  where
    co_var = mkWildCoVar (mkCoKind s t)

mkCoPredCo :: Coercion -> Coercion -> Coercion -> Coercion 
-- Creates a coercion between (s1~t1) => r1  and (s2~t2) => r2 
mkCoPredCo = mkCoPredTy 


splitCoPredTy_maybe :: Type -> Maybe (Type, Type, Type)
splitCoPredTy_maybe ty
  | Just (cv,r) <- splitForAllTy_maybe ty
  , isCoVar cv
  , Just (s,t) <- coVarKind_maybe cv
  = Just (s,t,r)
  | otherwise
  = Nothing

-- | Tests whether a type is just a type equality predicate
isEqPredTy :: Type -> Bool
isEqPredTy (PredTy pred) = isEqPred pred
isEqPredTy _             = False

-- | Creates a type equality predicate
mkEqPred :: (Type, Type) -> PredType
mkEqPred (ty1, ty2) = EqPred ty1 ty2

-- | Splits apart a type equality predicate, if the supplied 'PredType' is one.
-- Panics otherwise
getEqPredTys :: PredType -> (Type,Type)
getEqPredTys (EqPred ty1 ty2) = (ty1, ty2)
getEqPredTys other	      = pprPanic "getEqPredTys" (ppr other)

isIdentityCoercion :: Coercion -> Bool
isIdentityCoercion co  
  = case coercionKind co of
       (t1,t2) -> t1 `coreEqType` t2
\end{code}

%************************************************************************
%*									*
            Building coercions
%*									*
%************************************************************************

Coercion kind and type mk's (make saturated TyConApp CoercionTyCon{...} args)

\begin{code}
-- | Make a coercion from the specified coercion 'TyCon' and the 'Type' arguments to
-- that coercion. Try to use the @mk*Coercion@ family of functions instead of using this function
-- if possible
mkCoercion :: TyCon -> [Type] -> Coercion
mkCoercion coCon args = ASSERT( tyConArity coCon == length args ) 
                        TyConApp coCon args

mkCoVarCoercion :: CoVar -> Coercion 
mkCoVarCoercion cv = mkTyVarTy cv 

-- | Apply a 'Coercion' to another 'Coercion', which is presumably a
-- 'Coercion' constructor of some kind
mkAppCoercion :: Coercion -> Coercion -> Coercion
mkAppCoercion co1 co2 = mkAppTy co1 co2

-- | Applies multiple 'Coercion's to another 'Coercion', from left to right.
-- See also 'mkAppCoercion'
mkAppsCoercion :: Coercion -> [Coercion] -> Coercion
mkAppsCoercion co1 tys = foldl mkAppTy co1 tys

-- | Apply a type constructor to a list of coercions.
mkTyConCoercion :: TyCon -> [Coercion] -> Coercion
mkTyConCoercion con cos = mkTyConApp con cos

-- | Make a function 'Coercion' between two other 'Coercion's
mkFunCoercion :: Coercion -> Coercion -> Coercion
mkFunCoercion co1 co2 = mkFunTy co1 co2 -- NB: Handles correctly the forall for eqpreds!

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


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

mkSymCoercion :: Coercion -> Coercion
-- ^ 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@.
mkSymCoercion g = mkCoercion symCoercionTyCon [g]

mkTransCoercion :: Coercion -> Coercion -> Coercion
-- ^ Create a new 'Coercion' by exploiting transitivity on the two given 'Coercion's.
mkTransCoercion g1 g2 = mkCoercion transCoercionTyCon [g1, g2]

mkLeftCoercion :: Coercion -> Coercion
-- ^ From an application 'Coercion' build a 'Coercion' that asserts the equality of 
-- the "functions" on either side of the type equality. So if @c@ has kind @f x ~ g y@ then:
--
-- > mkLeftCoercion c :: f ~ g
mkLeftCoercion co = mkCoercion leftCoercionTyCon [co]

mkRightCoercion :: Coercion -> Coercion
-- ^ From an application 'Coercion' build a 'Coercion' that asserts the equality of 
-- the "arguments" on either side of the type equality. So if @c@ has kind @f x ~ g y@ then:
--
-- > mkLeftCoercion c :: x ~ y
mkRightCoercion co = mkCoercion rightCoercionTyCon [co]

mkCsel1Coercion, mkCsel2Coercion, mkCselRCoercion :: Coercion -> Coercion
mkCsel1Coercion co = mkCoercion csel1CoercionTyCon [co]
mkCsel2Coercion co = mkCoercion csel2CoercionTyCon [co]
mkCselRCoercion co = mkCoercion cselRCoercionTyCon [co]

-------------------------------
mkInstCoercion :: Coercion -> Type -> Coercion
-- ^ Instantiates a 'Coercion' with a 'Type' argument. If possible, it immediately performs
-- the resulting beta-reduction, otherwise it creates a suspended instantiation.
mkInstCoercion co ty = mkCoercion instCoercionTyCon  [co, ty]

mkInstsCoercion :: Coercion -> [Type] -> Coercion
-- ^ As 'mkInstCoercion', but instantiates the coercion with a number of type arguments, left-to-right
mkInstsCoercion co tys = foldl mkInstCoercion co tys

-- | Manufacture a coercion from this air. Needless to say, this is not usually safe,
-- but it is used when we know we are dealing with bottom, which is one case in which 
-- it is safe.  This is also used implement the @unsafeCoerce#@ primitive.
-- Optimise by pushing down through type constructors
mkUnsafeCoercion :: Type -> Type -> Coercion
mkUnsafeCoercion (TyConApp tc1 tys1) (TyConApp tc2 tys2)
  | tc1 == tc2
  = TyConApp tc1 (zipWith mkUnsafeCoercion tys1 tys2)

mkUnsafeCoercion (FunTy a1 r1) (FunTy a2 r2)
  = FunTy (mkUnsafeCoercion a1 a2) (mkUnsafeCoercion r1 r2)

mkUnsafeCoercion ty1 ty2 
  | ty1 `coreEqType` ty2 = ty1
  | otherwise            = mkCoercion unsafeCoercionTyCon [ty1, ty2]

-- See note [Newtype coercions] in TyCon

-- | Create a coercion suitable for the given 'TyCon'. The 'Name' should be that of a
-- new coercion 'TyCon', 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.
mkNewTypeCoercion :: Name -> TyCon -> [TyVar] -> Type -> TyCon
mkNewTypeCoercion name tycon tvs rhs_ty
  = mkCoercionTyCon name arity desc
  where
    arity = length tvs
    desc = CoAxiom { co_ax_tvs = tvs 
                   , co_ax_lhs = mkTyConApp tycon (mkTyVarTys tvs)
                   , co_ax_rhs = rhs_ty }

-- | Create a coercion identifying a @data@, @newtype@ or @type@ representation type
-- and its family instance.  It has the form @Co tvs :: F ts ~ R tvs@, where @Co@ is 
-- the coercion tycon built here, @F@ the family tycon and @R@ the (derived)
-- representation tycon.
mkFamInstCoercion :: Name	-- ^ Unique name for the coercion tycon
		  -> [TyVar]	-- ^ Type parameters of the coercion (@tvs@)
		  -> TyCon	-- ^ Family tycon (@F@)
		  -> [Type]	-- ^ Type instance (@ts@)
		  -> TyCon	-- ^ Representation tycon (@R@)
		  -> TyCon	-- ^ Coercion tycon (@Co@)
mkFamInstCoercion name tvs family inst_tys rep_tycon
  = mkCoercionTyCon name arity desc
  where
    arity = length tvs
    desc = CoAxiom { co_ax_tvs = tvs
                   , co_ax_lhs = mkTyConApp family inst_tys 
                   , co_ax_rhs = mkTyConApp rep_tycon (mkTyVarTys tvs) }


mkClassPPredCo :: Class -> [Coercion] -> Coercion
mkClassPPredCo cls = (PredTy . ClassP cls)

mkIParamPredCo :: (IPName Name) -> Coercion -> Coercion
mkIParamPredCo ipn = (PredTy . IParam ipn)

mkEqPredCo :: Coercion -> Coercion -> Coercion 
mkEqPredCo co1 co2 = PredTy (EqPred co1 co2)


\end{code}


%************************************************************************
%*									*
            Coercion Type Constructors
%*									*
%************************************************************************

Example.  The coercion ((sym c) (sym d) (sym e))
will be represented by (TyConApp sym [c, sym d, sym e])
If sym c :: p1=q1
   sym d :: p2=q2
   sym e :: p3=q3
then ((sym c) (sym d) (sym e)) :: (p1 p2 p3)=(q1 q2 q3)

\begin{code}
-- | Coercion type constructors: avoid using these directly and instead use 
-- the @mk*Coercion@ and @split*Coercion@ family of functions if possible.
--
-- Each coercion TyCon is built with the special CoercionTyCon record and
-- carries its own kinding rule.  Such CoercionTyCons must be fully applied
-- by any TyConApp in which they are applied, however they may also be over
-- applied (see example above) and the kinding function must deal with this.
symCoercionTyCon, transCoercionTyCon, leftCoercionTyCon, 
  rightCoercionTyCon, instCoercionTyCon, unsafeCoercionTyCon,
  csel1CoercionTyCon, csel2CoercionTyCon, cselRCoercionTyCon :: TyCon

symCoercionTyCon    = mkCoercionTyCon symCoercionTyConName   1 CoSym
transCoercionTyCon  = mkCoercionTyCon transCoercionTyConName 2 CoTrans
leftCoercionTyCon   = mkCoercionTyCon leftCoercionTyConName  1 CoLeft
rightCoercionTyCon  = mkCoercionTyCon rightCoercionTyConName 1 CoRight
instCoercionTyCon   =  mkCoercionTyCon instCoercionTyConName 2 CoInst
csel1CoercionTyCon  = mkCoercionTyCon csel1CoercionTyConName 1 CoCsel1
csel2CoercionTyCon  = mkCoercionTyCon csel2CoercionTyConName 1 CoCsel2
cselRCoercionTyCon  = mkCoercionTyCon cselRCoercionTyConName 1 CoCselR
unsafeCoercionTyCon = mkCoercionTyCon unsafeCoercionTyConName 2 CoUnsafe

transCoercionTyConName, symCoercionTyConName, leftCoercionTyConName, 
   rightCoercionTyConName, instCoercionTyConName, unsafeCoercionTyConName,
   csel1CoercionTyConName, csel2CoercionTyConName, cselRCoercionTyConName :: Name

transCoercionTyConName 	= mkCoConName (fsLit "trans") transCoercionTyConKey transCoercionTyCon
symCoercionTyConName   	= mkCoConName (fsLit "sym") symCoercionTyConKey symCoercionTyCon
leftCoercionTyConName  	= mkCoConName (fsLit "left") leftCoercionTyConKey leftCoercionTyCon
rightCoercionTyConName 	= mkCoConName (fsLit "right") rightCoercionTyConKey rightCoercionTyCon
instCoercionTyConName  	= mkCoConName (fsLit "inst") instCoercionTyConKey instCoercionTyCon
csel1CoercionTyConName  = mkCoConName (fsLit "csel1") csel1CoercionTyConKey csel1CoercionTyCon
csel2CoercionTyConName  = mkCoConName (fsLit "csel2") csel2CoercionTyConKey csel2CoercionTyCon
cselRCoercionTyConName  = mkCoConName (fsLit "cselR") cselRCoercionTyConKey cselRCoercionTyCon
unsafeCoercionTyConName = mkCoConName (fsLit "CoUnsafe") unsafeCoercionTyConKey unsafeCoercionTyCon

mkCoConName :: FastString -> Unique -> TyCon -> Name
mkCoConName occ key coCon = mkWiredInName gHC_PRIM (mkTcOccFS occ)
                            key (ATyCon coCon) BuiltInSyntax
\end{code}

\begin{code}
------------
decompLR_maybe :: (Type,Type) -> Maybe ((Type,Type), (Type,Type))
-- Helper for left and right.  Finds coercion kind of its input and
-- returns the left and right projections of the coercion...
--
-- if c :: t1 s1 ~ t2 s2 then splitCoercionKindOf c = ((t1, t2), (s1, s2))
decompLR_maybe (ty1,ty2)
  | Just (ty_fun1, ty_arg1) <- splitAppTy_maybe ty1
  , Just (ty_fun2, ty_arg2) <- splitAppTy_maybe ty2
  = Just ((ty_fun1, ty_fun2),(ty_arg1, ty_arg2))
decompLR_maybe _ = Nothing

------------
decompInst_maybe :: (Type, Type) -> Maybe ((TyVar,TyVar), (Type,Type))
decompInst_maybe (ty1, ty2)
  | Just (tv1,r1) <- splitForAllTy_maybe ty1
  , Just (tv2,r2) <- splitForAllTy_maybe ty2
  = Just ((tv1,tv2), (r1,r2))
decompInst_maybe _ = Nothing

------------
decompCsel_maybe :: (Type, Type) -> Maybe ((Type,Type), (Type,Type), (Type,Type))
--   If         co :: (s1~t1 => r1) ~ (s2~t2 => r2)
-- Then   csel1 co ::            s1 ~ s2
--        csel2 co :: 		 t1 ~ t2
--        cselR co :: 		 r1 ~ r2
decompCsel_maybe (ty1, ty2)
  | Just (s1, t1, r1) <- splitCoPredTy_maybe ty1
  , Just (s2, t2, r2) <- splitCoPredTy_maybe ty2
  = Just ((s1,s2), (t1,t2), (r1,r2))
decompCsel_maybe _ = Nothing
\end{code}


%************************************************************************
%*									*
            Newtypes
%*									*
%************************************************************************

\begin{code}
instNewTyCon_maybe :: TyCon -> [Type] -> Maybe (Type, CoercionI)
-- ^ If @co :: T ts ~ rep_ty@ then:
--
-- > instNewTyCon_maybe T ts = Just (rep_ty, co)
instNewTyCon_maybe tc tys
  | Just (tvs, ty, mb_co_tc) <- unwrapNewTyCon_maybe tc
  = ASSERT( tys `lengthIs` tyConArity tc )
    Just (substTyWith tvs tys ty, 
	  case mb_co_tc of
	     Nothing    -> IdCo (mkTyConApp tc    tys)
	     Just co_tc -> ACo  (mkTyConApp co_tc tys))
  | otherwise
  = Nothing

-- this is here to avoid module loops
splitNewTypeRepCo_maybe :: Type -> Maybe (Type, Coercion)  
-- ^ Sometimes we want to look through a @newtype@ and get its associated coercion.
-- This function only strips *one layer* of @newtype@ off, so the caller will usually call
-- itself recursively. Furthermore, this function should only be applied to types of kind @*@,
-- hence the newtype is always saturated. If @co : ty ~ ty'@ then:
--
-- > splitNewTypeRepCo_maybe ty = Just (ty', co)
--
-- The function returns @Nothing@ for non-@newtypes@ or fully-transparent @newtype@s.
splitNewTypeRepCo_maybe ty 
  | Just ty' <- coreView ty = splitNewTypeRepCo_maybe ty'
splitNewTypeRepCo_maybe (TyConApp tc tys)
  | Just (ty', coi) <- instNewTyCon_maybe tc tys
  = case coi of
	ACo co -> Just (ty', co)
	IdCo _ -> panic "splitNewTypeRepCo_maybe"
			-- This case handled by coreView
splitNewTypeRepCo_maybe _
  = Nothing

-- | Determines syntactic equality of coercions
coreEqCoercion :: Coercion -> Coercion -> Bool
coreEqCoercion = coreEqType

coreEqCoercion2 :: RnEnv2 -> Coercion -> Coercion -> Bool
coreEqCoercion2 = coreEqType2
\end{code}


%************************************************************************
%*									*
            CoercionI and its constructors
%*									*
%************************************************************************

--------------------------------------
-- CoercionI smart constructors
--	lifted smart constructors of ordinary coercions

\begin{code}
-- | 'CoercionI' represents a /lifted/ ordinary 'Coercion', in that it
-- can represent either one of:
--
-- 1. A proper 'Coercion'
--
-- 2. The identity coercion
data CoercionI = IdCo Type | ACo Coercion

liftCoI :: (Type -> Type) -> CoercionI -> CoercionI
liftCoI f (IdCo ty) = IdCo (f ty)
liftCoI f (ACo ty)  = ACo (f ty)

liftCoI2 :: (Type -> Type -> Type) -> CoercionI -> CoercionI -> CoercionI
liftCoI2 f (IdCo ty1) (IdCo ty2) = IdCo (f ty1 ty2)
liftCoI2 f coi1       coi2       = ACo (f (fromCoI coi1) (fromCoI coi2))

liftCoIs :: ([Type] -> Type) -> [CoercionI] -> CoercionI
liftCoIs f cois = go_id [] cois
  where
    go_id rev_tys []               = IdCo (f (reverse rev_tys))
    go_id rev_tys (IdCo ty : cois) = go_id  (ty:rev_tys) cois
    go_id rev_tys (ACo  co : cois) = go_aco (co:rev_tys) cois

    go_aco rev_tys []               = ACo (f (reverse rev_tys))
    go_aco rev_tys (IdCo ty : cois) = go_aco (ty:rev_tys) cois
    go_aco rev_tys (ACo  co : cois) = go_aco (co:rev_tys) cois

instance Outputable CoercionI where
  ppr (IdCo _) = ptext (sLit "IdCo")
  ppr (ACo co) = ppr co

isIdentityCoI :: CoercionI -> Bool
isIdentityCoI (IdCo _) = True
isIdentityCoI (ACo _)  = False

-- | Return either the 'Coercion' contained within the 'CoercionI' or the given
-- 'Type' if the 'CoercionI' is the identity 'Coercion'
fromCoI :: CoercionI -> Type
fromCoI (IdCo ty) = ty	-- Identity coercion represented 
fromCoI (ACo co)  = co	-- 	by the type itself

-- | Smart constructor for @sym@ on 'CoercionI', see also 'mkSymCoercion'
mkSymCoI :: CoercionI -> CoercionI
mkSymCoI (IdCo ty) = IdCo ty
mkSymCoI (ACo co)  = ACo $ mkCoercion symCoercionTyCon [co] 
				-- the smart constructor
				-- is too smart with tyvars

-- | Smart constructor for @trans@ on 'CoercionI', see also 'mkTransCoercion'
mkTransCoI :: CoercionI -> CoercionI -> CoercionI
mkTransCoI (IdCo _) aco = aco
mkTransCoI aco (IdCo _) = aco
mkTransCoI (ACo co1) (ACo co2) = ACo $ mkTransCoercion co1 co2

-- | Smart constructor for type constructor application on 'CoercionI', see also 'mkAppCoercion'
mkTyConAppCoI :: TyCon -> [CoercionI] -> CoercionI
mkTyConAppCoI tyCon cois = liftCoIs (mkTyConApp tyCon) cois

-- | Smart constructor for honest-to-god 'Coercion' application on 'CoercionI', see also 'mkAppCoercion'
mkAppTyCoI :: CoercionI -> CoercionI -> CoercionI
mkAppTyCoI = liftCoI2 mkAppTy

mkFunTyCoI :: CoercionI -> CoercionI -> CoercionI
mkFunTyCoI = liftCoI2 mkFunTy

-- | Smart constructor for quantified 'Coercion's on 'CoercionI', see also 'mkForAllCoercion'
mkForAllTyCoI :: TyVar -> CoercionI -> CoercionI
mkForAllTyCoI tv = liftCoI (ForAllTy tv)

-- | Smart constructor for class 'Coercion's on 'CoercionI'. Satisfies:
--
-- > mkClassPPredCoI cls tys cois :: PredTy (cls tys) ~ PredTy (cls (tys `cast` cois))
mkClassPPredCoI :: Class -> [CoercionI] -> CoercionI
mkClassPPredCoI cls = liftCoIs (PredTy . ClassP cls)

-- | Smart constructor for implicit parameter 'Coercion's on 'CoercionI'. Similar to 'mkClassPPredCoI'
mkIParamPredCoI :: (IPName Name) -> CoercionI -> CoercionI 
mkIParamPredCoI ipn = liftCoI (PredTy . IParam ipn)

-- | Smart constructor for type equality 'Coercion's on 'CoercionI'. Similar to 'mkClassPPredCoI'
mkEqPredCoI :: CoercionI -> CoercionI -> CoercionI
mkEqPredCoI = liftCoI2 (\t1 t2 -> PredTy (EqPred t1 t2))

mkCoPredCoI :: CoercionI -> CoercionI -> CoercionI -> CoercionI 
mkCoPredCoI coi1 coi2 coi3 =   mkFunTyCoI (mkEqPredCoI coi1 coi2) coi3


\end{code}

%************************************************************************
%*									*
	     The kind of a type, and of a coercion
%*									*
%************************************************************************

\begin{code}
typeKind :: Type -> Kind
typeKind ty@(TyConApp tc tys) 
  | isCoercionTyCon tc = typeKind (fst (coercionKind ty))
  | otherwise          = kindAppResult (tyConKind tc) tys
	-- During coercion optimisation we *do* match a type
	-- against a coercion (see OptCoercion.matchesAxiomLhs)
	-- So the use of typeKind in Unify.match_kind must work on coercions too
	-- Hence the isCoercionTyCon case above

typeKind (PredTy pred)	      = predKind pred
typeKind (AppTy fun _)        = kindFunResult (typeKind fun)
typeKind (ForAllTy _ ty)      = typeKind ty
typeKind (TyVarTy tyvar)      = tyVarKind tyvar
typeKind (FunTy _arg res)
    -- Hack alert.  The kind of (Int -> Int#) is liftedTypeKind (*), 
    --              not unliftedTypKind (#)
    -- The only things that can be after a function arrow are
    --   (a) types (of kind openTypeKind or its sub-kinds)
    --   (b) kinds (of super-kind TY) (e.g. * -> (* -> *))
    | isTySuperKind k         = k
    | otherwise               = ASSERT( isSubOpenTypeKind k) liftedTypeKind 
    where
      k = typeKind res

------------------
predKind :: PredType -> Kind
predKind (EqPred {}) = coSuperKind	-- A coercion kind!
predKind (ClassP {}) = liftedTypeKind	-- Class and implicitPredicates are
predKind (IParam {}) = liftedTypeKind 	-- always represented by lifted types

------------------
-- | If it is the case that
--
-- > c :: (t1 ~ t2)
--
-- i.e. the kind of @c@ is a 'CoercionKind' relating @t1@ and @t2@, 
-- then @coercionKind c = (t1, t2)@.
coercionKind :: Coercion -> (Type, Type)
coercionKind ty@(TyVarTy a) | isCoVar a = coVarKind a
                            | otherwise = (ty, ty)
coercionKind (AppTy ty1 ty2) 
  = let (s1, t1) = coercionKind ty1
        (s2, t2) = coercionKind ty2 in
    (mkAppTy s1 s2, mkAppTy t1 t2)
coercionKind co@(TyConApp tc args)
  | Just (ar, desc) <- isCoercionTyCon_maybe tc 
    -- CoercionTyCons carry their kinding rule, so we use it here
  = WARN( not (length args >= ar), ppr co )	-- Always saturated
    (let (ty1,  ty2)  = coTyConAppKind desc (take ar args)
	 (tys1, tys2) = coercionKinds (drop ar args)
     in (mkAppTys ty1 tys1, mkAppTys ty2 tys2))

  | otherwise
  = let (lArgs, rArgs) = coercionKinds args in
    (TyConApp tc lArgs, TyConApp tc rArgs)

coercionKind (FunTy ty1 ty2) 
  = let (t1, t2) = coercionKind ty1
        (s1, s2) = coercionKind ty2 in
    (mkFunTy t1 s1, mkFunTy t2 s2)

coercionKind (ForAllTy tv ty)
  | isCoVar tv
--     c1 :: s1~s2  c2 :: t1~t2   c3 :: r1~r2
--    ----------------------------------------------
--    c1~c2 => c3  ::  (s1~t1) => r1 ~ (s2~t2) => r2
--      or
--    forall (_:c1~c2)
  = let (c1,c2) = coVarKind tv
    	(s1,s2) = coercionKind c1
    	(t1,t2) = coercionKind c2
    	(r1,r2) = coercionKind ty
    in
    (mkCoPredTy s1 t1 r1, mkCoPredTy s2 t2 r2)

  | otherwise
--     c1 :: s1~s2  c2 :: t1~t2   c3 :: r1~r2
--   ----------------------------------------------
--    forall a:k. c :: forall a:k. t1 ~ forall a:k. t2
  = let (ty1, ty2) = coercionKind ty in
    (ForAllTy tv ty1, ForAllTy tv ty2)

coercionKind (PredTy (ClassP cl args)) 
  = let (lArgs, rArgs) = coercionKinds args in
    (PredTy (ClassP cl lArgs), PredTy (ClassP cl rArgs))
coercionKind (PredTy (IParam name ty))
  = let (ty1, ty2) = coercionKind ty in
    (PredTy (IParam name ty1), PredTy (IParam name ty2))
coercionKind (PredTy (EqPred c1 c2)) 
  = pprTrace "coercionKind" (pprEqPred (c1,c2)) $
  -- These should not show up in coercions at all
  -- becuase they are in the form of for-alls
    let k1 = coercionKindPredTy c1
        k2 = coercionKindPredTy c2 in
    (k1,k2)
  where
    coercionKindPredTy c = let (t1, t2) = coercionKind c in mkCoKind t1 t2

------------------
-- | Apply 'coercionKind' to multiple 'Coercion's
coercionKinds :: [Coercion] -> ([Type], [Type])
coercionKinds tys = unzip $ map coercionKind tys

------------------
-- | 'coTyConAppKind' is given a list of the type arguments to the 'CoTyCon',
-- and constructs the types that the resulting coercion relates.
-- Fails (in the monad) if ill-kinded.
-- Typically the monad is 
--   either the Lint monad (with the consistency-check flag = True), 
--   or the ID monad with a panic on failure (and the consistency-check flag = False)
coTyConAppKind 
    :: CoTyConDesc
    -> [Type]	  		-- Exactly right number of args
    -> (Type, Type)		-- Kind of this application
coTyConAppKind CoUnsafe (ty1:ty2:_)
  = (ty1,ty2)
coTyConAppKind CoSym (co:_) 
  | (ty1,ty2) <- coercionKind co = (ty2,ty1)
coTyConAppKind CoTrans (co1:co2:_) 
  = (fst (coercionKind co1), snd (coercionKind co2))
coTyConAppKind CoLeft (co:_) 
  | Just (res,_) <- decompLR_maybe (coercionKind co) = res
coTyConAppKind CoRight (co:_) 
  | Just (_,res) <- decompLR_maybe (coercionKind co) = res
coTyConAppKind CoCsel1 (co:_) 
  | Just (res,_,_) <- decompCsel_maybe (coercionKind co) = res
coTyConAppKind CoCsel2 (co:_) 
  | Just (_,res,_) <- decompCsel_maybe (coercionKind co) = res
coTyConAppKind CoCselR (co:_) 
  | Just (_,_,res) <- decompCsel_maybe (coercionKind co) = res
coTyConAppKind CoInst (co:ty:_) 
  | Just ((tv1,tv2), (ty1,ty2)) <- decompInst_maybe (coercionKind co)
  = (substTyWith [tv1] [ty] ty1, substTyWith [tv2] [ty] ty2) 
coTyConAppKind (CoAxiom { co_ax_tvs = tvs 
                        , co_ax_lhs = lhs_ty, co_ax_rhs = rhs_ty }) cos
  = (substTyWith tvs tys1 lhs_ty, substTyWith tvs tys2 rhs_ty)
  where
    (tys1, tys2) = coercionKinds cos
coTyConAppKind desc cos = pprTrace "coTyConAppKind" (ppr desc $$ braces (vcat 
                             [ ppr co <+> dcolon <+> pprEqPred (coercionKind co)
                             | co <- cos ])) $
                          coercionKind (head cos)
\end{code}