Type.lhs 51.5 KB
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% (c) The University of Glasgow 2006
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% (c) The GRASP/AQUA Project, Glasgow University, 1998
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Type - public interface
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\begin{code}
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{-# OPTIONS -w #-}
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-- The above warning supression flag is a temporary kludge.
-- While working on this module you are encouraged to remove it and fix
-- any warnings in the module. See
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--     http://hackage.haskell.org/trac/ghc/wiki/Commentary/CodingStyle#Warnings
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-- for details

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module Type (
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        -- re-exports from TypeRep
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	TyThing(..), Type, PredType(..), ThetaType, 
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	funTyCon,
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	-- Kinds
        Kind, SimpleKind, KindVar,
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        kindFunResult, splitKindFunTys, splitKindFunTysN,
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        liftedTypeKindTyCon, openTypeKindTyCon, unliftedTypeKindTyCon,
        argTypeKindTyCon, ubxTupleKindTyCon,

        liftedTypeKind, unliftedTypeKind, openTypeKind,
        argTypeKind, ubxTupleKind,

        tySuperKind, coSuperKind, 

        isLiftedTypeKind, isUnliftedTypeKind, isOpenTypeKind,
        isUbxTupleKind, isArgTypeKind, isKind, isTySuperKind, 
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        isCoSuperKind, isSuperKind, isCoercionKind, isEqPred,
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	mkArrowKind, mkArrowKinds,

        isSubArgTypeKind, isSubOpenTypeKind, isSubKind, defaultKind, eqKind,
        isSubKindCon,
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	-- Re-exports from TyCon
	PrimRep(..),
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	mkTyVarTy, mkTyVarTys, getTyVar, getTyVar_maybe, isTyVarTy,

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	mkAppTy, mkAppTys, splitAppTy, splitAppTys, 
	splitAppTy_maybe, repSplitAppTy_maybe,
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	mkFunTy, mkFunTys, splitFunTy, splitFunTy_maybe, 
	splitFunTys, splitFunTysN,
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	funResultTy, funArgTy, zipFunTys, isFunTy,
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	mkTyConApp, mkTyConTy, 
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	tyConAppTyCon, tyConAppArgs, 
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	splitTyConApp_maybe, splitTyConApp, 
        splitNewTyConApp_maybe, splitNewTyConApp,
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	repType, repType', typePrimRep, coreView, tcView, kindView,
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	mkForAllTy, mkForAllTys, splitForAllTy_maybe, splitForAllTys, 
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	applyTy, applyTys, isForAllTy, dropForAlls,
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	-- Source types
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	predTypeRep, mkPredTy, mkPredTys, pprSourceTyCon, mkFamilyTyConApp,
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	-- Newtypes
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	newTyConInstRhs,
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	-- Lifting and boxity
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	isUnLiftedType, isUnboxedTupleType, isAlgType, isPrimitiveType,
	isStrictType, isStrictPred, 
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	-- Free variables
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	tyVarsOfType, tyVarsOfTypes, tyVarsOfPred, tyVarsOfTheta,
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	typeKind, addFreeTyVars,
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	-- Tidying up for printing
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	tidyType,      tidyTypes,
	tidyOpenType,  tidyOpenTypes,
	tidyTyVarBndr, tidyFreeTyVars,
	tidyOpenTyVar, tidyOpenTyVars,
	tidyTopType,   tidyPred,
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	tidyKind,
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	-- Comparison
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	coreEqType, tcEqType, tcEqTypes, tcCmpType, tcCmpTypes, 
	tcEqPred, tcCmpPred, tcEqTypeX, 
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	-- Seq
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	seqType, seqTypes,
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	-- Type substitutions
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	TvSubstEnv, emptyTvSubstEnv,	-- Representation widely visible
	TvSubst(..), emptyTvSubst,	-- Representation visible to a few friends
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	mkTvSubst, mkOpenTvSubst, zipOpenTvSubst, zipTopTvSubst, mkTopTvSubst, notElemTvSubst,
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	getTvSubstEnv, setTvSubstEnv, getTvInScope, extendTvInScope,
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 	extendTvSubst, extendTvSubstList, isInScope, composeTvSubst, zipTyEnv,
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        isEmptyTvSubst,
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	-- Performing substitution on types
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	substTy, substTys, substTyWith, substTheta, 
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	substPred, substTyVar, substTyVars, substTyVarBndr, deShadowTy, lookupTyVar,
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	-- Pretty-printing
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	pprType, pprParendType, pprTypeApp, pprTyThingCategory, pprForAll,
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	pprPred, pprTheta, pprThetaArrow, pprClassPred, pprKind, pprParendKind
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    ) where
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#include "HsVersions.h"

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-- We import the representation and primitive functions from TypeRep.
-- Many things are reexported, but not the representation!

import TypeRep

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-- friends:
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import Var
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import VarEnv
import VarSet

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import Name
import Class
import PrelNames
import TyCon
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-- others
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import StaticFlags
import Util
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import Outputable
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import UniqSet
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import Data.List
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import Data.Maybe	( isJust )
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\end{code}

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%************************************************************************
%*									*
		Type representation
%*									*
%************************************************************************

In Core, we "look through" non-recursive newtypes and PredTypes.

\begin{code}
{-# INLINE coreView #-}
coreView :: Type -> Maybe Type
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-- Strips off the *top layer only* of a type to give 
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-- its underlying representation type. 
-- Returns Nothing if there is nothing to look through.
--
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-- In the case of newtypes, it returns
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--	*either* a vanilla TyConApp (recursive newtype, or non-saturated)
--	*or*     the newtype representation (otherwise), meaning the
--			type written in the RHS of the newtype decl,
--			which may itself be a newtype
--
-- Example: newtype R = MkR S
--	    newtype S = MkS T
--	    newtype T = MkT (T -> T)
--   expandNewTcApp on R gives Just S
--	            on S gives Just T
--		    on T gives Nothing	 (no expansion)

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-- By being non-recursive and inlined, this case analysis gets efficiently
-- joined onto the case analysis that the caller is already doing
coreView (NoteTy _ ty) 	   = Just ty
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coreView (PredTy p)
  | isEqPred p             = Nothing
  | otherwise    	   = Just (predTypeRep p)
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coreView (TyConApp tc tys) | Just (tenv, rhs, tys') <- coreExpandTyCon_maybe tc tys 
			   = Just (mkAppTys (substTy (mkTopTvSubst tenv) rhs) tys')
				-- Its important to use mkAppTys, rather than (foldl AppTy),
				-- because the function part might well return a 
				-- partially-applied type constructor; indeed, usually will!
coreView ty		   = Nothing

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-----------------------------------------------
{-# INLINE tcView #-}
tcView :: Type -> Maybe Type
-- Same, but for the type checker, which just looks through synonyms
tcView (NoteTy _ ty) 	 = Just ty
tcView (TyConApp tc tys) | Just (tenv, rhs, tys') <- tcExpandTyCon_maybe tc tys 
			 = Just (mkAppTys (substTy (mkTopTvSubst tenv) rhs) tys')
tcView ty		 = Nothing
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-----------------------------------------------
{-# INLINE kindView #-}
kindView :: Kind -> Maybe Kind
-- C.f. coreView, tcView
-- For the moment, we don't even handle synonyms in kinds
kindView (NoteTy _ k) = Just k
kindView other	      = Nothing
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\end{code}


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%************************************************************************
%*									*
\subsection{Constructor-specific functions}
%*									*
%************************************************************************
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---------------------------------------------------------------------
				TyVarTy
				~~~~~~~
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\begin{code}
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mkTyVarTy  :: TyVar   -> Type
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mkTyVarTy  = TyVarTy
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mkTyVarTys :: [TyVar] -> [Type]
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mkTyVarTys = map mkTyVarTy -- a common use of mkTyVarTy
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getTyVar :: String -> Type -> TyVar
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getTyVar msg ty = case getTyVar_maybe ty of
		    Just tv -> tv
		    Nothing -> panic ("getTyVar: " ++ msg)
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isTyVarTy :: Type -> Bool
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isTyVarTy ty = isJust (getTyVar_maybe ty)

getTyVar_maybe :: Type -> Maybe TyVar
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getTyVar_maybe ty | Just ty' <- coreView ty = getTyVar_maybe ty'
getTyVar_maybe (TyVarTy tv) 	 	    = Just tv  
getTyVar_maybe other	         	    = Nothing
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\end{code}


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---------------------------------------------------------------------
				AppTy
				~~~~~
We need to be pretty careful with AppTy to make sure we obey the 
invariant that a TyConApp is always visibly so.  mkAppTy maintains the
invariant: use it.
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\begin{code}
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mkAppTy orig_ty1 orig_ty2
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  = mk_app orig_ty1
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  where
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    mk_app (NoteTy _ ty1)    = mk_app ty1
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    mk_app (TyConApp tc tys) = mkTyConApp tc (tys ++ [orig_ty2])
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    mk_app ty1		     = AppTy orig_ty1 orig_ty2
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	-- Note that the TyConApp could be an 
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	-- under-saturated type synonym.  GHC allows that; e.g.
	--	type Foo k = k a -> k a
	--	type Id x = x
	--	foo :: Foo Id -> Foo Id
	--
	-- Here Id is partially applied in the type sig for Foo,
	-- but once the type synonyms are expanded all is well
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mkAppTys :: Type -> [Type] -> Type
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mkAppTys orig_ty1 []	    = orig_ty1
	-- This check for an empty list of type arguments
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	-- avoids the needless loss of a type synonym constructor.
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	-- For example: mkAppTys Rational []
	--   returns to (Ratio Integer), which has needlessly lost
	--   the Rational part.
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mkAppTys orig_ty1 orig_tys2
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  = mk_app orig_ty1
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  where
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    mk_app (NoteTy _ ty1)    = mk_app ty1
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    mk_app (TyConApp tc tys) = mkTyConApp tc (tys ++ orig_tys2)
				-- mkTyConApp: see notes with mkAppTy
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    mk_app ty1		     = foldl AppTy orig_ty1 orig_tys2
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-------------
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splitAppTy_maybe :: Type -> Maybe (Type, Type)
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splitAppTy_maybe ty | Just ty' <- coreView ty
		    = splitAppTy_maybe ty'
splitAppTy_maybe ty = repSplitAppTy_maybe ty
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-------------
repSplitAppTy_maybe :: Type -> Maybe (Type,Type)
-- Does the AppTy split, but assumes that any view stuff is already done
repSplitAppTy_maybe (FunTy ty1 ty2)   = Just (TyConApp funTyCon [ty1], ty2)
repSplitAppTy_maybe (AppTy ty1 ty2)   = Just (ty1, ty2)
repSplitAppTy_maybe (TyConApp tc tys) = case snocView tys of
						Just (tys', ty') -> Just (TyConApp tc tys', ty')
						Nothing		 -> Nothing
repSplitAppTy_maybe other = Nothing
-------------
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splitAppTy :: Type -> (Type, Type)
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splitAppTy ty = case splitAppTy_maybe ty of
			Just pr -> pr
			Nothing -> panic "splitAppTy"
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-------------
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splitAppTys :: Type -> (Type, [Type])
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splitAppTys ty = split ty ty []
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  where
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    split orig_ty ty args | Just ty' <- coreView ty = split orig_ty ty' args
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    split orig_ty (AppTy ty arg)        args = split ty ty (arg:args)
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    split orig_ty (TyConApp tc tc_args) args = (TyConApp tc [], tc_args ++ args)
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    split orig_ty (FunTy ty1 ty2)       args = ASSERT( null args )
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					       (TyConApp funTyCon [], [ty1,ty2])
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    split orig_ty ty		        args = (orig_ty, args)
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\end{code}

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---------------------------------------------------------------------
				FunTy
				~~~~~

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\begin{code}
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mkFunTy :: Type -> Type -> Type
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mkFunTy (PredTy (EqPred ty1 ty2)) res = mkForAllTy (mkWildCoVar (PredTy (EqPred ty1 ty2))) res
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mkFunTy arg res = FunTy arg res
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mkFunTys :: [Type] -> Type -> Type
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mkFunTys tys ty = foldr mkFunTy ty tys
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isFunTy :: Type -> Bool 
isFunTy ty = isJust (splitFunTy_maybe ty)

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splitFunTy :: Type -> (Type, Type)
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splitFunTy ty | Just ty' <- coreView ty = splitFunTy ty'
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splitFunTy (FunTy arg res)   = (arg, res)
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splitFunTy other	     = pprPanic "splitFunTy" (ppr other)
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splitFunTy_maybe :: Type -> Maybe (Type, Type)
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splitFunTy_maybe ty | Just ty' <- coreView ty = splitFunTy_maybe ty'
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splitFunTy_maybe (FunTy arg res)   = Just (arg, res)
splitFunTy_maybe other	           = Nothing
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splitFunTys :: Type -> ([Type], Type)
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splitFunTys ty = split [] ty ty
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  where
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    split args orig_ty ty | Just ty' <- coreView ty = split args orig_ty ty'
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    split args orig_ty (FunTy arg res) 	 = split (arg:args) res res
    split args orig_ty ty                = (reverse args, orig_ty)
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splitFunTysN :: Int -> Type -> ([Type], Type)
-- Split off exactly n arg tys
splitFunTysN 0 ty = ([], ty)
splitFunTysN n ty = case splitFunTy ty of { (arg, res) ->
		    case splitFunTysN (n-1) res of { (args, res) ->
		    (arg:args, res) }}

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zipFunTys :: Outputable a => [a] -> Type -> ([(a,Type)], Type)
zipFunTys orig_xs orig_ty = split [] orig_xs orig_ty orig_ty
  where
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    split acc []     nty ty  	           = (reverse acc, nty)
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    split acc xs     nty ty 
	  | Just ty' <- coreView ty 	   = split acc xs nty ty'
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    split acc (x:xs) nty (FunTy arg res)   = split ((x,arg):acc) xs res res
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    split acc (x:xs) nty ty                = pprPanic "zipFunTys" (ppr orig_xs <+> ppr orig_ty)
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funResultTy :: Type -> Type
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funResultTy ty | Just ty' <- coreView ty = funResultTy ty'
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funResultTy (FunTy arg res)   = res
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funResultTy ty		      = pprPanic "funResultTy" (ppr ty)
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funArgTy :: Type -> Type
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funArgTy ty | Just ty' <- coreView ty = funArgTy ty'
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funArgTy (FunTy arg res)   = arg
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funArgTy ty		   = pprPanic "funArgTy" (ppr ty)
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\end{code}


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---------------------------------------------------------------------
				TyConApp
				~~~~~~~~
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@mkTyConApp@ is a key function, because it builds a TyConApp, FunTy or PredTy,
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as apppropriate.
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\begin{code}
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mkTyConApp :: TyCon -> [Type] -> Type
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mkTyConApp tycon tys
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  | isFunTyCon tycon, [ty1,ty2] <- tys
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  = FunTy ty1 ty2
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  | otherwise
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  = TyConApp tycon tys
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mkTyConTy :: TyCon -> Type
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mkTyConTy tycon = mkTyConApp tycon []
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-- splitTyConApp "looks through" synonyms, because they don't
-- mean a distinct type, but all other type-constructor applications
-- including functions are returned as Just ..

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tyConAppTyCon :: Type -> TyCon
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tyConAppTyCon ty = fst (splitTyConApp ty)
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tyConAppArgs :: Type -> [Type]
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tyConAppArgs ty = snd (splitTyConApp ty)
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splitTyConApp :: Type -> (TyCon, [Type])
splitTyConApp ty = case splitTyConApp_maybe ty of
			Just stuff -> stuff
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			Nothing	   -> pprPanic "splitTyConApp" (ppr ty)
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splitTyConApp_maybe :: Type -> Maybe (TyCon, [Type])
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splitTyConApp_maybe ty | Just ty' <- coreView ty = splitTyConApp_maybe ty'
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splitTyConApp_maybe (TyConApp tc tys) = Just (tc, tys)
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splitTyConApp_maybe (FunTy arg res)   = Just (funTyCon, [arg,res])
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splitTyConApp_maybe other	      = Nothing
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-- Sometimes we do NOT want to look throught a newtype.  When case matching
-- on a newtype we want a convenient way to access the arguments of a newty
-- constructor so as to properly form a coercion.
splitNewTyConApp :: Type -> (TyCon, [Type])
splitNewTyConApp ty = case splitNewTyConApp_maybe ty of
			Just stuff -> stuff
			Nothing	   -> pprPanic "splitNewTyConApp" (ppr ty)
splitNewTyConApp_maybe :: Type -> Maybe (TyCon, [Type])
splitNewTyConApp_maybe ty | Just ty' <- tcView ty = splitNewTyConApp_maybe ty'
splitNewTyConApp_maybe (TyConApp tc tys) = Just (tc, tys)
splitNewTyConApp_maybe (FunTy arg res)   = Just (funTyCon, [arg,res])
splitNewTyConApp_maybe other	      = Nothing

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newTyConInstRhs :: TyCon -> [Type] -> Type
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-- Unwrap one 'layer' of newtype
-- Use the eta'd version if possible
newTyConInstRhs tycon tys 
    = ASSERT2( equalLength tvs tys1, ppr tycon $$ ppr tys $$ ppr tvs )
      mkAppTys (substTyWith tvs tys1 ty) tys2
  where
    (tvs, ty)    = newTyConEtadRhs tycon
    (tys1, tys2) = splitAtList tvs tys
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\end{code}
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---------------------------------------------------------------------
				SynTy
				~~~~~

Notes on type synonyms
~~~~~~~~~~~~~~~~~~~~~~
The various "split" functions (splitFunTy, splitRhoTy, splitForAllTy) try
to return type synonyms whereever possible. Thus
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	type Foo a = a -> a

we want 
	splitFunTys (a -> Foo a) = ([a], Foo a)
not			           ([a], a -> a)

The reason is that we then get better (shorter) type signatures in 
interfaces.  Notably this plays a role in tcTySigs in TcBinds.lhs.
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		Representation types
		~~~~~~~~~~~~~~~~~~~~
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repType looks through 
	(a) for-alls, and
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	(b) synonyms
	(c) predicates
	(d) usage annotations
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	(e) all newtypes, including recursive ones, but not newtype families
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It's useful in the back end.
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\begin{code}
repType :: Type -> Type
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-- Only applied to types of kind *; hence tycons are saturated
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repType ty | Just ty' <- coreView ty = repType ty'
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repType (ForAllTy _ ty)  = repType ty
repType (TyConApp tc tys)
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  | isNewTyCon tc
  , (tvs, rep_ty) <- newTyConRep tc
  = -- Recursive newtypes are opaque to coreView
    -- but we must expand them here.  Sure to
    -- be saturated because repType is only applied
    -- to types of kind *
    ASSERT( tys `lengthIs` tyConArity tc )
    repType (substTyWith tvs tys rep_ty)

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repType ty = ty

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-- repType' aims to be a more thorough version of repType
-- For now it simply looks through the TyConApp args too
repType' ty -- | pprTrace "repType'" (ppr ty $$ ppr (go1 ty)) False = undefined
            | otherwise = go1 ty 
 where 
        go1 = go . repType
        go (TyConApp tc tys) = mkTyConApp tc (map repType' tys)
        go ty = ty


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-- ToDo: this could be moved to the code generator, using splitTyConApp instead
-- of inspecting the type directly.
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typePrimRep :: Type -> PrimRep
typePrimRep ty = case repType ty of
		   TyConApp tc _ -> tyConPrimRep tc
		   FunTy _ _	 -> PtrRep
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		   AppTy _ _	 -> PtrRep	-- See note below
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		   TyVarTy _	 -> PtrRep
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		   other	 -> pprPanic "typePrimRep" (ppr ty)
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	-- Types of the form 'f a' must be of kind *, not *#, so
	-- we are guaranteed that they are represented by pointers.
	-- The reason is that f must have kind *->*, not *->*#, because
	-- (we claim) there is no way to constrain f's kind any other
	-- way.
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\end{code}


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---------------------------------------------------------------------
				ForAllTy
				~~~~~~~~
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\begin{code}
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mkForAllTy :: TyVar -> Type -> Type
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mkForAllTy tyvar ty
  = mkForAllTys [tyvar] ty
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mkForAllTys :: [TyVar] -> Type -> Type
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mkForAllTys tyvars ty = foldr ForAllTy ty tyvars
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isForAllTy :: Type -> Bool
isForAllTy (NoteTy _ ty)  = isForAllTy ty
isForAllTy (ForAllTy _ _) = True
isForAllTy other_ty	  = False
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splitForAllTy_maybe :: Type -> Maybe (TyVar, Type)
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splitForAllTy_maybe ty = splitFAT_m ty
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  where
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    splitFAT_m ty | Just ty' <- coreView ty = splitFAT_m ty'
    splitFAT_m (ForAllTy tyvar ty)	    = Just(tyvar, ty)
    splitFAT_m _			    = Nothing
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splitForAllTys :: Type -> ([TyVar], Type)
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splitForAllTys ty = split ty ty []
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   where
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     split orig_ty ty tvs | Just ty' <- coreView ty = split orig_ty ty' tvs
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     split orig_ty (ForAllTy tv ty)  tvs = split ty ty (tv:tvs)
     split orig_ty t		     tvs = (reverse tvs, orig_ty)
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dropForAlls :: Type -> Type
dropForAlls ty = snd (splitForAllTys ty)
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\end{code}

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-- (mkPiType now in CoreUtils)
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applyTy, applyTys
~~~~~~~~~~~~~~~~~
Instantiate a for-all type with one or more type arguments.
Used when we have a polymorphic function applied to type args:
	f t1 t2
Then we use (applyTys type-of-f [t1,t2]) to compute the type of
the expression. 
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\begin{code}
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applyTy :: Type -> Type -> Type
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applyTy ty arg | Just ty' <- coreView ty = applyTy ty' arg
applyTy (ForAllTy tv ty) arg = substTyWith [tv] [arg] ty
applyTy other		 arg = panic "applyTy"
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applyTys :: Type -> [Type] -> Type
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-- This function is interesting because 
--	a) the function may have more for-alls than there are args
--	b) less obviously, it may have fewer for-alls
-- For case (b) think of 
--	applyTys (forall a.a) [forall b.b, Int]
-- This really can happen, via dressing up polymorphic types with newtype
-- clothing.  Here's an example:
--	newtype R = R (forall a. a->a)
--	foo = case undefined :: R of
--		R f -> f ()

applyTys orig_fun_ty []      = orig_fun_ty
applyTys orig_fun_ty arg_tys 
  | n_tvs == n_args 	-- The vastly common case
  = substTyWith tvs arg_tys rho_ty
  | n_tvs > n_args 	-- Too many for-alls
  = substTyWith (take n_args tvs) arg_tys 
		(mkForAllTys (drop n_args tvs) rho_ty)
  | otherwise		-- Too many type args
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  = ASSERT2( n_tvs > 0, ppr orig_fun_ty )	-- Zero case gives infnite loop!
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    applyTys (substTyWith tvs (take n_tvs arg_tys) rho_ty)
	     (drop n_tvs arg_tys)
  where
    (tvs, rho_ty) = splitForAllTys orig_fun_ty 
    n_tvs = length tvs
    n_args = length arg_tys     
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\end{code}
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%************************************************************************
%*									*
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\subsection{Source types}
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%*									*
%************************************************************************
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A "source type" is a type that is a separate type as far as the type checker is
concerned, but which has low-level representation as far as the back end is concerned.
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Source types are always lifted.
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The key function is predTypeRep which gives the representation of a source type:
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\begin{code}
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mkPredTy :: PredType -> Type
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mkPredTy pred = PredTy pred
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mkPredTys :: ThetaType -> [Type]
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mkPredTys preds = map PredTy preds

predTypeRep :: PredType -> Type
-- Convert a PredType to its "representation type";
-- the post-type-checking type used by all the Core passes of GHC.
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-- Unwraps only the outermost level; for example, the result might
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-- be a newtype application
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predTypeRep (IParam _ ty)     = ty
predTypeRep (ClassP clas tys) = mkTyConApp (classTyCon clas) tys
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	-- Result might be a newtype application, but the consumer will
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	-- look through that too if necessary
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predTypeRep (EqPred ty1 ty2) = pprPanic "predTypeRep" (ppr (EqPred ty1 ty2))
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mkFamilyTyConApp :: TyCon -> [Type] -> Type
-- Given a family instance TyCon and its arg types, return the
-- corresponding family type.  E.g.
--	data family T a
--	data instance T (Maybe b) = MkT b	-- Instance tycon :RTL
-- Then 
--	mkFamilyTyConApp :RTL Int  =  T (Maybe Int)
mkFamilyTyConApp tc tys
  | Just (fam_tc, fam_tys) <- tyConFamInst_maybe tc
  , let fam_subst = zipTopTvSubst (tyConTyVars tc) tys
  = mkTyConApp fam_tc (substTys fam_subst fam_tys)
  | otherwise
  = mkTyConApp tc tys

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-- Pretty prints a tycon, using the family instance in case of a
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-- representation tycon.  For example
--  	e.g.  data T [a] = ...
-- In that case we want to print `T [a]', where T is the family TyCon
pprSourceTyCon tycon 
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  | Just (fam_tc, tys) <- tyConFamInst_maybe tycon
  = ppr $ fam_tc `TyConApp` tys	       -- can't be FunTyCon
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  | otherwise
  = ppr tycon
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\end{code}
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%************************************************************************
%*									*
\subsection{Kinds and free variables}
%*									*
%************************************************************************

---------------------------------------------------------------------
		Finding the kind of a type
		~~~~~~~~~~~~~~~~~~~~~~~~~~
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\begin{code}
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typeKind :: Type -> Kind
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typeKind (TyConApp tycon tys) = ASSERT( not (isCoercionTyCon tycon) )
				   -- We should be looking for the coercion kind,
				   -- not the type kind
				foldr (\_ k -> kindFunResult k) (tyConKind tycon) tys
typeKind (NoteTy _ ty)	      = typeKind ty
typeKind (PredTy pred)	      = predKind pred
typeKind (AppTy fun arg)      = kindFunResult (typeKind fun)
typeKind (ForAllTy tv 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
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\end{code}


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---------------------------------------------------------------------
		Free variables of a type
		~~~~~~~~~~~~~~~~~~~~~~~~
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\begin{code}
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tyVarsOfType :: Type -> TyVarSet
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-- NB: for type synonyms tyVarsOfType does *not* expand the synonym
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tyVarsOfType (TyVarTy tv)		= unitVarSet tv
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tyVarsOfType (TyConApp tycon tys)	= tyVarsOfTypes tys
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tyVarsOfType (NoteTy (FTVNote tvs) ty2) = tvs
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tyVarsOfType (PredTy sty)		= tyVarsOfPred sty
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tyVarsOfType (FunTy arg res)		= tyVarsOfType arg `unionVarSet` tyVarsOfType res
tyVarsOfType (AppTy fun arg)		= tyVarsOfType fun `unionVarSet` tyVarsOfType arg
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tyVarsOfType (ForAllTy tyvar ty)	= delVarSet (tyVarsOfType ty) tyvar
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tyVarsOfTypes :: [Type] -> TyVarSet
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tyVarsOfTypes tys = foldr (unionVarSet.tyVarsOfType) emptyVarSet tys

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tyVarsOfPred :: PredType -> TyVarSet
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tyVarsOfPred (IParam _ ty)    = tyVarsOfType ty
tyVarsOfPred (ClassP _ tys)   = tyVarsOfTypes tys
tyVarsOfPred (EqPred ty1 ty2) = tyVarsOfType ty1 `unionVarSet` tyVarsOfType ty2
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tyVarsOfTheta :: ThetaType -> TyVarSet
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tyVarsOfTheta = foldr (unionVarSet . tyVarsOfPred) emptyVarSet
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-- Add a Note with the free tyvars to the top of the type
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addFreeTyVars :: Type -> Type
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addFreeTyVars ty@(NoteTy (FTVNote _) _)      = ty
addFreeTyVars ty			     = NoteTy (FTVNote (tyVarsOfType ty)) ty
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\end{code}
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%************************************************************************
%*									*
\subsection{TidyType}
%*									*
%************************************************************************
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tidyTy tidies up a type for printing in an error message, or in
an interface file.
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It doesn't change the uniques at all, just the print names.
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\begin{code}
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tidyTyVarBndr :: TidyEnv -> TyVar -> (TidyEnv, TyVar)
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tidyTyVarBndr env@(tidy_env, subst) tyvar
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  = case tidyOccName tidy_env (getOccName name) of
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      (tidy', occ') -> ((tidy', subst'), tyvar'')
	where
	  subst' = extendVarEnv subst tyvar tyvar''
	  tyvar' = setTyVarName tyvar name'
	  name'  = tidyNameOcc name occ'
		-- Don't forget to tidy the kind for coercions!
	  tyvar'' | isCoVar tyvar = setTyVarKind tyvar' kind'
		  | otherwise	  = tyvar'
	  kind'  = tidyType env (tyVarKind tyvar)
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  where
    name = tyVarName tyvar
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tidyFreeTyVars :: TidyEnv -> TyVarSet -> TidyEnv
-- Add the free tyvars to the env in tidy form,
-- so that we can tidy the type they are free in
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tidyFreeTyVars env tyvars = fst (tidyOpenTyVars env (varSetElems tyvars))

tidyOpenTyVars :: TidyEnv -> [TyVar] -> (TidyEnv, [TyVar])
tidyOpenTyVars env tyvars = mapAccumL tidyOpenTyVar env tyvars

tidyOpenTyVar :: TidyEnv -> TyVar -> (TidyEnv, TyVar)
-- Treat a new tyvar as a binder, and give it a fresh tidy name
tidyOpenTyVar env@(tidy_env, subst) tyvar
  = case lookupVarEnv subst tyvar of
	Just tyvar' -> (env, tyvar')		-- Already substituted
	Nothing	    -> tidyTyVarBndr env tyvar	-- Treat it as a binder
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tidyType :: TidyEnv -> Type -> Type
tidyType env@(tidy_env, subst) ty
  = go ty
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  where
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    go (TyVarTy tv)	    = case lookupVarEnv subst tv of
				Nothing  -> TyVarTy tv
				Just tv' -> TyVarTy tv'
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    go (TyConApp tycon tys) = let args = map go tys
			      in args `seqList` TyConApp tycon args
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    go (NoteTy note ty)     = (NoteTy $! (go_note note)) $! (go ty)
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    go (PredTy sty)	    = PredTy (tidyPred env sty)
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    go (AppTy fun arg)	    = (AppTy $! (go fun)) $! (go arg)
    go (FunTy fun arg)	    = (FunTy $! (go fun)) $! (go arg)
    go (ForAllTy tv ty)	    = ForAllTy tvp $! (tidyType envp ty)
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			      where
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			        (envp, tvp) = tidyTyVarBndr env tv
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    go_note note@(FTVNote ftvs) = note	-- No need to tidy the free tyvars

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tidyTypes env tys = map (tidyType env) tys
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tidyPred :: TidyEnv -> PredType -> PredType
tidyPred env (IParam n ty)     = IParam n (tidyType env ty)
tidyPred env (ClassP clas tys) = ClassP clas (tidyTypes env tys)
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tidyPred env (EqPred ty1 ty2)  = EqPred (tidyType env ty1) (tidyType env ty2)
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\end{code}


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@tidyOpenType@ grabs the free type variables, tidies them
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and then uses @tidyType@ to work over the type itself

\begin{code}
tidyOpenType :: TidyEnv -> Type -> (TidyEnv, Type)
tidyOpenType env ty
  = (env', tidyType env' ty)
  where
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tidyOpenTypes :: TidyEnv -> [Type] -> (TidyEnv, [Type])
tidyOpenTypes env tys = mapAccumL tidyOpenType env tys

tidyTopType :: Type -> Type
tidyTopType ty = tidyType emptyTidyEnv ty
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\end{code}

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\begin{code}
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tidyKind :: TidyEnv -> Kind -> (TidyEnv, Kind)
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tidyKind env k = tidyOpenType env k
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\end{code}

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%************************************************************************
%*									*
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\subsection{Liftedness}
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%*									*
%************************************************************************

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\begin{code}
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isUnLiftedType :: Type -> Bool
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	-- isUnLiftedType returns True for forall'd unlifted types:
	--	x :: forall a. Int#
	-- I found bindings like these were getting floated to the top level.
	-- They are pretty bogus types, mind you.  It would be better never to
	-- construct them

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isUnLiftedType ty | Just ty' <- coreView ty = isUnLiftedType ty'
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isUnLiftedType (ForAllTy tv ty)  = isUnLiftedType ty
isUnLiftedType (TyConApp tc _)   = isUnLiftedTyCon tc
isUnLiftedType other		 = False	
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isUnboxedTupleType :: Type -> Bool
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isUnboxedTupleType ty = case splitTyConApp_maybe ty of
			   Just (tc, ty_args) -> isUnboxedTupleTyCon tc
			   other	      -> False
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-- Should only be applied to *types*; hence the assert
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isAlgType :: Type -> Bool
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isAlgType ty = case splitTyConApp_maybe ty of
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			Just (tc, ty_args) -> ASSERT( ty_args `lengthIs` tyConArity tc )
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					      isAlgTyCon tc
			other		   -> False
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\end{code}

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@isStrictType@ computes whether an argument (or let RHS) should
be computed strictly or lazily, based only on its type.
Works just like isUnLiftedType, except that it has a special case 
for dictionaries.  Since it takes account of ClassP, you might think
this function should be in TcType, but isStrictType is used by DataCon,
which is below TcType in the hierarchy, so it's convenient to put it here.

\begin{code}
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isStrictType (PredTy pred)     = isStrictPred pred
isStrictType ty | Just ty' <- coreView ty = isStrictType ty'
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isStrictType (ForAllTy tv ty)  = isStrictType ty
isStrictType (TyConApp tc _)   = isUnLiftedTyCon tc
isStrictType other	       = False	

isStrictPred (ClassP clas _) = opt_DictsStrict && not (isNewTyCon (classTyCon clas))
isStrictPred other	     = False
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	-- We may be strict in dictionary types, but only if it 
	-- has more than one component.
	-- [Being strict in a single-component dictionary risks
	--  poking the dictionary component, which is wrong.]
\end{code}

\begin{code}
isPrimitiveType :: Type -> Bool
-- Returns types that are opaque to Haskell.
-- Most of these are unlifted, but now that we interact with .NET, we
-- may have primtive (foreign-imported) types that are lifted
isPrimitiveType ty = case splitTyConApp_maybe ty of
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			Just (tc, ty_args) -> ASSERT( ty_args `lengthIs` tyConArity tc )
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					      isPrimTyCon tc
			other		   -> False
\end{code}

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%************************************************************************
%*									*
\subsection{Sequencing on types
%*									*
%************************************************************************

\begin{code}
seqType :: Type -> ()
seqType (TyVarTy tv) 	  = tv `seq` ()
seqType (AppTy t1 t2) 	  = seqType t1 `seq` seqType t2
seqType (FunTy t1 t2) 	  = seqType t1 `seq` seqType t2
seqType (NoteTy note t2)  = seqNote note `seq` seqType t2
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seqType (PredTy p) 	  = seqPred p
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seqType (TyConApp tc tys) = tc `seq` seqTypes tys
seqType (ForAllTy tv ty)  = tv `seq` seqType ty

seqTypes :: [Type] -> ()
seqTypes []       = ()
seqTypes (ty:tys) = seqType ty `seq` seqTypes tys

seqNote :: TyNote -> ()
seqNote (FTVNote set) = sizeUniqSet set `seq` ()
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seqPred :: PredType -> ()
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seqPred (ClassP c tys)   = c `seq` seqTypes tys
seqPred (IParam n ty)    = n `seq` seqType ty
seqPred (EqPred ty1 ty2) = seqType ty1 `seq` seqType ty2
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\end{code}


%************************************************************************
%*									*
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		Equality for Core types 
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	(We don't use instances so that we know where it happens)
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%*									*
%************************************************************************

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Note that eqType works right even for partial applications of newtypes.
See Note [Newtype eta] in TyCon.lhs

\begin{code}
coreEqType :: Type -> Type -> Bool
coreEqType t1 t2
  = eq rn_env t1 t2
  where
    rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfType t1 `unionVarSet` tyVarsOfType t2))

    eq env (TyVarTy tv1)       (TyVarTy tv2)     = rnOccL env tv1 == rnOccR env tv2
    eq env (ForAllTy tv1 t1)   (ForAllTy tv2 t2) = eq (rnBndr2 env tv1 tv2) t1 t2
    eq env (AppTy s1 t1)       (AppTy s2 t2)     = eq env s1 s2 && eq env t1 t2
    eq env (FunTy s1 t1)       (FunTy s2 t2)     = eq env s1 s2 && eq env t1 t2
    eq env (TyConApp tc1 tys1) (TyConApp tc2 tys2) 
	| tc1 == tc2, all2 (eq env) tys1 tys2 = True
			-- The lengths should be equal because
			-- the two types have the same kind
	-- NB: if the type constructors differ that does not 
	--     necessarily mean that the types aren't equal
	--     (synonyms, newtypes)
	-- Even if the type constructors are the same, but the arguments
	-- differ, the two types could be the same (e.g. if the arg is just
	-- ignored in the RHS).  In both these cases we fall through to an 
	-- attempt to expand one side or the other.

	-- Now deal with newtypes, synonyms, pred-tys
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    eq env t1 t2 | Just t1' <- coreView t1 = eq env t1' t2 
		 | Just t2' <- coreView t2 = eq env t1 t2' 
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	-- Fall through case; not equal!
    eq env t1 t2 = False
\end{code}
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%************************************************************************
%*									*
		Comparision for source types 
	(We don't use instances so that we know where it happens)
%*									*
%************************************************************************
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Note that 
	tcEqType, tcCmpType 
do *not* look through newtypes, PredTypes
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\begin{code}
tcEqType :: Type -> Type -> Bool
tcEqType t1 t2 = isEqual $ cmpType t1 t2

tcEqTypes :: [Type] -> [Type] -> Bool
tcEqTypes tys1 tys2 = isEqual $ cmpTypes tys1 tys2

tcCmpType :: Type -> Type -> Ordering
tcCmpType t1 t2 = cmpType t1 t2

tcCmpTypes :: [Type] -> [Type] -> Ordering
tcCmpTypes tys1 tys2 = cmpTypes tys1 tys2

tcEqPred :: PredType -> PredType -> Bool
tcEqPred p1 p2 = isEqual $ cmpPred p1 p2

tcCmpPred :: PredType -> PredType -> Ordering
tcCmpPred p1 p2 = cmpPred p1 p2

tcEqTypeX :: RnEnv2 -> Type -> Type -> Bool
tcEqTypeX env t1 t2 = isEqual $ cmpTypeX env t1 t2
\end{code}

Now here comes the real worker

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\begin{code}
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cmpType :: Type -> Type -> Ordering
cmpType t1 t2 = cmpTypeX rn_env t1 t2
  where
    rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfType t1 `unionVarSet` tyVarsOfType t2))

cmpTypes :: [Type] -> [Type] -> Ordering
cmpTypes ts1 ts2 = cmpTypesX rn_env ts1 ts2
  where
    rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfTypes ts1 `unionVarSet` tyVarsOfTypes ts2))

cmpPred :: PredType -> PredType -> Ordering
cmpPred p1 p2 = cmpPredX rn_env p1 p2
  where
    rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfPred p1 `unionVarSet` tyVarsOfPred p2))
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cmpTypeX :: RnEnv2 -> Type -> Type -> Ordering	-- Main workhorse
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cmpTypeX env t1 t2 | Just t1' <- tcView t1 = cmpTypeX env t1' t2
		   | Just t2' <- tcView t2 = cmpTypeX env t1 t2'
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cmpTypeX env (TyVarTy tv1)       (TyVarTy tv2)       = rnOccL env tv1 `compare` rnOccR env tv2
cmpTypeX env (ForAllTy tv1 t1)   (ForAllTy tv2 t2)   = cmpTypeX (rnBndr2 env tv1 tv2) t1 t2
cmpTypeX env (AppTy s1 t1)       (AppTy s2 t2)       = cmpTypeX env s1 s2 `thenCmp` cmpTypeX env t1 t2
cmpTypeX env (FunTy s1 t1)       (FunTy s2 t2)       = cmpTypeX env s1 s2 `thenCmp` cmpTypeX env t1 t2
cmpTypeX env (PredTy p1)         (PredTy p2)         = cmpPredX env p1 p2
cmpTypeX env (TyConApp tc1 tys1) (TyConApp tc2 tys2) = (tc1 `compare` tc2) `thenCmp` cmpTypesX env tys1 tys2
cmpTypeX env t1			(NoteTy _ t2)	     = cmpTypeX env t1 t2

    -- Deal with the rest: TyVarTy < AppTy < FunTy < TyConApp < ForAllTy < PredTy
cmpTypeX env (AppTy _ _) (TyVarTy _) = GT
    
cmpTypeX env (FunTy _ _) (TyVarTy _) = GT
cmpTypeX env (FunTy _ _) (AppTy _ _) = GT
    
cmpTypeX env (TyConApp _ _) (TyVarTy _) = GT
cmpTypeX env (TyConApp _ _) (AppTy _ _) = GT
cmpTypeX env (TyConApp _ _) (FunTy _ _) = GT
    
cmpTypeX env (ForAllTy _ _) (TyVarTy _)    = GT
cmpTypeX env (ForAllTy _ _) (AppTy _ _)    = GT
cmpTypeX env (ForAllTy _ _) (FunTy _ _)    = GT
cmpTypeX env (ForAllTy _ _) (TyConApp _ _) = GT

cmpTypeX env (PredTy _)   t2		= GT

cmpTypeX env _ _ = LT

-------------
cmpTypesX :: RnEnv2 -> [Type] -> [Type] -> Ordering
cmpTypesX env []        []        = EQ
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cmpTypesX env []        tys       = LT
cmpTypesX env ty        []        = GT

-------------
cmpPredX :: RnEnv2 -> PredType -> PredType -> Ordering
cmpPredX env (IParam n1 ty1) (IParam n2 ty2) = (n1 `compare` n2) `thenCmp` cmpTypeX env ty1 ty2
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	-- Compare names only for implicit parameters
	-- This comparison is used exclusively (I believe) 
	-- for the Avails finite map built in TcSimplify
	-- If the types differ we keep them distinct so that we see 
	-- a distinct pair to run improvement on 
cmpPredX env (ClassP c1 tys1) (ClassP c2 tys2) = (c1 `compare` c2) `thenCmp` (cmpTypesX env tys1 tys2)
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cmpPredX env (EqPred ty1 ty2) (EqPred ty1' ty2') = (cmpTypeX env ty1 ty1') `thenCmp` (cmpTypeX env ty2 ty2')
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-- Constructor order: IParam < ClassP < EqPred
cmpPredX env (IParam {})     _		    = LT
cmpPredX env (ClassP {})    (IParam {})     = GT
cmpPredX env (ClassP {})    (EqPred {})     = LT
cmpPredX env (EqPred {})    _		    = GT
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\end{code}

PredTypes are used as a FM key in TcSimplify, 
so we take the easy path and make them an instance of Ord

\begin{code}
instance Eq  PredType where { (==)    = tcEqPred }
instance Ord PredType where { compare = tcCmpPred }
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\end{code}

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%************************************************************************
%*									*
		Type substitutions
%*									*
%************************************************************************

\begin{code}
data TvSubst 		
  = TvSubst InScopeSet 	-- The in-scope type variables
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	    TvSubstEnv	-- The substitution itself
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	-- See Note [Apply Once]
	-- and Note [Extending the TvSubstEnv]
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{- ----------------------------------------------------------

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Note [Apply Once]
~~~~~~~~~~~~~~~~~
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We use TvSubsts to instantiate things, and we might instantiate
	forall a b. ty
\with the types
	[a, b], or [b, a].
So the substition might go [a->b, b->a].  A similar situation arises in Core
when we find a beta redex like
	(/\ a /\ b -> e) b a
Then we also end up with a substition that permutes type variables. Other
variations happen to; for example [a -> (a, b)].  

	***************************************************
	*** So a TvSubst must be applied precisely once ***
	***************************************************

A TvSubst is not idempotent, but, unlike the non-idempotent substitution
we use during unifications, it must not be repeatedly applied.
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Note [Extending the TvSubst]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The following invariant should hold of a TvSubst

	The in-scope set is needed *only* to
	guide the generation of fresh uniques

	In particular, the *kind* of the type variables in 
	the in-scope set is not relevant

This invariant allows a short-cut when the TvSubstEnv is empty:
if the TvSubstEnv is empty --- i.e. (isEmptyTvSubt subst) holds ---
then (substTy subst ty) does nothing.

For example, consider:
	(/\a. /\b:(a~Int). ...b..) Int
We substitute Int for 'a'.  The Unique of 'b' does not change, but
nevertheless we add 'b' to the TvSubstEnv, because b's type does change

This invariant has several crucial consequences:

* In substTyVarBndr, we need extend the TvSubstEnv 
	- if the unique has changed
	- or if the kind has changed

* In substTyVar, we do not need to consult the in-scope set;
  the TvSubstEnv is enough

* In substTy, substTheta, we can short-circuit when the TvSubstEnv is empty
  

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-------------------------------------------------------------- -}


type TvSubstEnv = TyVarEnv Type
	-- A TvSubstEnv is used both inside a TvSubst (with the apply-once
	-- invariant discussed in Note [Apply Once]), and also independently
	-- in the middle of matching, and unification (see Types.Unify)
	-- So you have to look at the context to know if it's idempotent or
	-- apply-once or whatever
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emptyTvSubstEnv :: TvSubstEnv
emptyTvSubstEnv = emptyVarEnv
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composeTvSubst :: InScopeSet -> TvSubstEnv -> TvSubstEnv -> TvSubstEnv
-- (compose env1 env2)(x) is env1(env2(x)); i.e. apply env2 then env1
-- It assumes that both are idempotent
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-- Typically, env1 is the refinement to a base substitution env2
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composeTvSubst in_scope env1 env2
  = env1 `plusVarEnv` mapVarEnv (substTy subst1) env2
	-- First apply env1 to the range of env2
	-- Then combine the two, making sure that env1 loses if
	-- both bind the same variable; that's why env1 is the
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	--  *left* argument to plusVarEnv, because the right arg wins
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  where
    subst1 = TvSubst in_scope env1

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emptyTvSubst = TvSubst emptyInScopeSet emptyVarEnv
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isEmptyTvSubst :: TvSubst -> Bool
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	 -- See Note [Extending the TvSubstEnv]
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isEmptyTvSubst (TvSubst _ env) = isEmptyVarEnv env

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mkTvSubst :: InScopeSet -> TvSubstEnv -> TvSubst
mkTvSubst = TvSubst

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getTvSubstEnv :: TvSubst -> TvSubstEnv
getTvSubstEnv (TvSubst _ env) = env

getTvInScope :: TvSubst -> InScopeSet
getTvInScope (TvSubst in_scope _) = in_scope

isInScope :: Var -> TvSubst -> Bool
isInScope v (TvSubst in_scope _) = v `elemInScopeSet` in_scope

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notElemTvSubst :: TyVar -> TvSubst -> Bool
notElemTvSubst tv (TvSubst _ env) = not (tv `elemVarEnv` env)

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setTvSubstEnv :: TvSubst -> TvSubstEnv -> TvSubst
setTvSubstEnv (TvSubst in_scope _) env = TvSubst in_scope env

extendTvInScope :: TvSubst -> [Var] -> TvSubst
extendTvInScope (TvSubst in_scope env) vars = TvSubst (extendInScopeSetList in_scope vars) env

extendTvSubst :: TvSubst -> TyVar -> Type -> TvSubst
extendTvSubst (TvSubst in_scope env) tv ty = TvSubst in_scope (extendVarEnv env tv ty)

extendTvSubstList :: TvSubst -> [TyVar] -> [Type] -> TvSubst
extendTvSubstList (TvSubst in_scope env) tvs tys 
  = TvSubst in_scope (extendVarEnvList env (tvs `zip` tys))

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-- mkOpenTvSubst and zipOpenTvSubst generate the in-scope set from
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-- the types given; but it's just a thunk so with a bit of luck
-- it'll never be evaluated

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mkOpenTvSubst :: TvSubstEnv -> TvSubst
mkOpenTvSubst env = TvSubst (mkInScopeSet (tyVarsOfTypes (varEnvElts env))) env
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zipOpenTvSubst :: [TyVar] -> [Type] -> TvSubst
zipOpenTvSubst tyvars tys 
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#ifdef DEBUG
  | length tyvars /= length tys
  = pprTrace "zipOpenTvSubst" (ppr tyvars $$ ppr tys) emptyTvSubst
  | otherwise
#endif
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  = TvSubst (mkInScopeSet (tyVarsOfTypes tys)) (zipTyEnv tyvars tys)

-- mkTopTvSubst is called when doing top-level substitutions.
-- Here we expect that the free vars of the range of the
-- substitution will be empty.
mkTopTvSubst :: [(TyVar, Type)] -> TvSubst
mkTopTvSubst prs = TvSubst emptyInScopeSet (mkVarEnv prs)

zipTopTvSubst :: [TyVar] -> [Type] -> TvSubst
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zipTopTvSubst tyvars tys 
#ifdef DEBUG
  | length tyvars /= length tys
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  = pprTrace "zipTopTvSubst" (ppr tyvars $$ ppr tys) emptyTvSubst