\begin{code}

    module TcCanonical(
        mkCanonical, mkCanonicals, mkCanonicalFEV, canWanteds, canGivens,
        canOccursCheck, canEq,
        rewriteWithFunDeps
     ) where

    #include "HsVersions.h"

    import BasicTypes
    import Type
    import TcRnTypes
    import FunDeps
    import qualified TcMType as TcM
    import TcType
    import TcErrors
    import Coercion
    import Class
    import TyCon
    import TypeRep
    import Name
    import Var
    import VarEnv		( TidyEnv )
    import Outputable
    import Control.Monad    ( unless, when, zipWithM, zipWithM_ )
    import MonadUtils
    import Control.Applicative ( (<|>) )

    import VarSet
    import Bag

    import HsBinds
    import TcSMonad
    import FastString

\end{code} Note [Canonicalisation] ~~~~~~~~~~~~~~~~~~~~~~~ * Converts (Constraint f) _which_does_not_contain_proper_implications_ to CanonicalCts * Unary: treats individual constraints one at a time * Does not do any zonking * Lives in TcS monad so that it can create new skolem variables %************************************************************************ %* * %* Flattening (eliminating all function symbols) * %* * %************************************************************************ Note [Flattening] ~~~~~~~~~~~~~~~~~~~~ flatten ty ==> (xi, cc) where xi has no type functions cc = Auxiliary given (equality) constraints constraining the fresh type variables in xi. Evidence for these is always the identity coercion, because internally the fresh flattening skolem variables are actually identified with the types they have been generated to stand in for. Note that it is flatten's job to flatten *every type function it sees*. flatten is only called on *arguments* to type functions, by canEqGiven. Recall that in comments we use alpha[flat = ty] to represent a flattening skolem variable alpha which has been generated to stand in for ty. ----- Example of flattening a constraint: ------ flatten (List (F (G Int))) ==> (xi, cc) where xi = List alpha cc = { G Int ~ beta[flat = G Int], F beta ~ alpha[flat = F beta] } Here * alpha and beta are 'flattening skolem variables'. * All the constraints in cc are 'given', and all their coercion terms are the identity. NB: Flattening Skolems only occur in canonical constraints, which are never zonked, so we don't need to worry about zonking doing accidental unflattening. Note that we prefer to leave type synonyms unexpanded when possible, so when the flattener encounters one, it first asks whether its transitive expansion contains any type function applications. If so, it expands the synonym and proceeds; if not, it simply returns the unexpanded synonym. TODO: caching the information about whether transitive synonym expansions contain any type function applications would speed things up a bit; right now we waste a lot of energy traversing the same types multiple times. \begin{code}

    -- Flatten a bunch of types all at once.
    flattenMany :: CtFlavor -> [Type] -> TcS ([Xi], [Coercion], CanonicalCts)
    -- Coercions :: Xi ~ Type
    flattenMany ctxt tys
      = do { (xis, cos, cts_s) <- mapAndUnzip3M (flatten ctxt) tys
           ; return (xis, cos, andCCans cts_s) }

    -- Flatten a type to get rid of type function applications, returning
    -- the new type-function-free type, and a collection of new equality
    -- constraints.  See Note [Flattening] for more detail.
    flatten :: CtFlavor -> TcType -> TcS (Xi, Coercion, CanonicalCts)
    -- Postcondition: Coercion :: Xi ~ TcType
    flatten ctxt ty
      | Just ty' <- tcView ty
      = do { (xi, co, ccs) <- flatten ctxt ty'
    	-- Preserve type synonyms if possible
    	-- We can tell if ty' is function-free by
    	-- whether there are any floated constraints
           ; if isEmptyCCan ccs then
                 return (ty, ty, emptyCCan)
             else
                 return (xi, co, ccs) }

    flatten _ v@(TyVarTy _)
      = return (v, v, emptyCCan)

    flatten ctxt (AppTy ty1 ty2)
      = do { (xi1,co1,c1) <- flatten ctxt ty1
           ; (xi2,co2,c2) <- flatten ctxt ty2
           ; return (mkAppTy xi1 xi2, mkAppCoercion co1 co2, c1 `andCCan` c2) }

    flatten ctxt (FunTy ty1 ty2)
      = do { (xi1,co1,c1) <- flatten ctxt ty1
           ; (xi2,co2,c2) <- flatten ctxt ty2
           ; return (mkFunTy xi1 xi2, mkFunCoercion co1 co2, c1 `andCCan` c2) }

    flatten fl (TyConApp tc tys)
      -- For a normal type constructor or data family application, we just
      -- recursively flatten the arguments.
      | not (isSynFamilyTyCon tc)
        = do { (xis,cos,ccs) <- flattenMany fl tys
             ; return (mkTyConApp tc xis, mkTyConCoercion tc cos, ccs) }

      -- Otherwise, it's a type function application, and we have to
      -- flatten it away as well, and generate a new given equality constraint
      -- between the application and a newly generated flattening skolem variable.
      | otherwise
      = ASSERT( tyConArity tc <= length tys )	-- Type functions are saturated
          do { (xis, cos, ccs) <- flattenMany fl tys
             ; let (xi_args, xi_rest)  = splitAt (tyConArity tc) xis
                   (cos_args, cos_rest) = splitAt (tyConArity tc) cos
    	       	 -- The type function might be *over* saturated
    		 -- in which case the remaining arguments should
    		 -- be dealt with by AppTys
                   fam_ty = mkTyConApp tc xi_args
                   fam_co = fam_ty -- identity

             ; (ret_co, rhs_var, ct) <-
                 if isGiven fl then
                   do { rhs_var <- newFlattenSkolemTy fam_ty
                      ; cv <- newGivenCoVar fam_ty rhs_var fam_co
                      ; let ct = CFunEqCan { cc_id     = cv
                                           , cc_flavor = fl -- Given
                                           , cc_fun    = tc
                                           , cc_tyargs = xi_args
                                           , cc_rhs    = rhs_var }
                      ; return $ (mkCoVarCoercion cv, rhs_var, ct) }
                 else -- Derived or Wanted: make a new *unification* flatten variable
                   do { rhs_var <- newFlexiTcSTy (typeKind fam_ty)
                      ; cv <- newCoVar fam_ty rhs_var
                      ; let ct = CFunEqCan { cc_id = cv
                                           , cc_flavor = mkWantedFlavor fl
                                               -- Always Wanted, not Derived
                                           , cc_fun = tc
                                           , cc_tyargs = xi_args
                                           , cc_rhs    = rhs_var }
                      ; return $ (mkCoVarCoercion cv, rhs_var, ct) }

             ; return ( foldl AppTy rhs_var xi_rest
                      , foldl AppTy (mkSymCoercion ret_co
                                        `mkTransCoercion` mkTyConCoercion tc cos_args) cos_rest
                      , ccs `extendCCans` ct) }

    flatten ctxt (PredTy pred)
      = do { (pred', co, ccs) <- flattenPred ctxt pred
           ; return (PredTy pred', co, ccs) }

    flatten ctxt ty@(ForAllTy {})
    -- We allow for-alls when, but only when, no type function
    -- applications inside the forall involve the bound type variables
    -- TODO: What if it is a (t1 ~ t2) => t3
    --       Must revisit when the New Coercion API is here!
      = do { let (tvs, rho) = splitForAllTys ty
           ; (rho', co, ccs) <- flatten ctxt rho
           ; let bad_eqs  = filterBag is_bad ccs
                 is_bad c = tyVarsOfCanonical c `intersectsVarSet` tv_set
                 tv_set   = mkVarSet tvs
           ; unless (isEmptyBag bad_eqs)
                    (flattenForAllErrorTcS ctxt ty bad_eqs)
           ; return (mkForAllTys tvs rho', mkForAllTys tvs co, ccs)  }

    ---------------
    flattenPred :: CtFlavor -> TcPredType -> TcS (TcPredType, Coercion, CanonicalCts)
    flattenPred ctxt (ClassP cls tys)
      = do { (tys', cos, ccs) <- flattenMany ctxt tys
           ; return (ClassP cls tys', mkClassPPredCo cls cos, ccs) }
    flattenPred ctxt (IParam nm ty)
      = do { (ty', co, ccs) <- flatten ctxt ty
           ; return (IParam nm ty', mkIParamPredCo nm co, ccs) }
    -- TODO: Handling of coercions between EqPreds must be revisited once the New Coercion API is ready!
    flattenPred ctxt (EqPred ty1 ty2)
      = do { (ty1', co1, ccs1) <- flatten ctxt ty1
           ; (ty2', co2, ccs2) <- flatten ctxt ty2
           ; return (EqPred ty1' ty2', mkEqPredCo co1 co2, ccs1 `andCCan` ccs2) }

\end{code} %************************************************************************ %* * %* Canonicalising given constraints * %* * %************************************************************************ \begin{code}

    canWanteds :: [WantedEvVar] -> TcS CanonicalCts
    canWanteds = fmap andCCans . mapM (\(EvVarX ev loc) -> mkCanonical (Wanted loc) ev)

    canGivens :: GivenLoc -> [EvVar] -> TcS CanonicalCts
    canGivens loc givens = do { ccs <- mapM (mkCanonical (Given loc)) givens
                              ; return (andCCans ccs) }

    mkCanonicals :: CtFlavor -> [EvVar] -> TcS CanonicalCts
    mkCanonicals fl vs = fmap andCCans (mapM (mkCanonical fl) vs)

    mkCanonicalFEV :: FlavoredEvVar -> TcS CanonicalCts
    mkCanonicalFEV (EvVarX ev fl) = mkCanonical fl ev

    mkCanonical :: CtFlavor -> EvVar -> TcS CanonicalCts
    mkCanonical fl ev = case evVarPred ev of
                            ClassP clas tys -> canClass fl ev clas tys
                            IParam ip ty    -> canIP    fl ev ip ty
                            EqPred ty1 ty2  -> canEq    fl ev ty1 ty2

    canClass :: CtFlavor -> EvVar -> Class -> [TcType] -> TcS CanonicalCts
    canClass fl v cn tys
      = do { (xis,cos,ccs) <- flattenMany fl tys  -- cos :: xis ~ tys
           ; let no_flattening_happened = isEmptyCCan ccs
                 dict_co = mkTyConCoercion (classTyCon cn) cos
           ; v_new <- if no_flattening_happened then return v
                      else if isGiven fl        then return v
                             -- The cos are all identities if fl=Given,
                             -- hence nothing to do
                      else do { v' <- newDictVar cn xis  -- D xis
                              ; when (isWanted fl) $ setDictBind v  (EvCast v' dict_co)
                              ; when (isGiven fl)  $ setDictBind v' (EvCast v (mkSymCoercion dict_co))
                                     -- NB: No more setting evidence for derived now
                              ; return v' }

           -- Add the superclasses of this one here, See Note [Adding superclasses].
           -- But only if we are not simplifying the LHS of a rule.
           ; sctx <- getTcSContext
           ; sc_cts <- if simplEqsOnly sctx then return emptyCCan
                       else newSCWorkFromFlavored v_new fl cn xis

           ; return (sc_cts `andCCan` ccs `extendCCans` CDictCan { cc_id     = v_new
                                                                 , cc_flavor = fl
                                                                 , cc_class  = cn
                                                                 , cc_tyargs = xis }) }

\end{code} Note [Adding superclasses] ~~~~~~~~~~~~~~~~~~~~~~~~~~ Since dictionaries are canonicalized only once in their lifetime, the place to add their superclasses is canonicalisation (The alternative would be to do it during constraint solving, but we'd have to be extremely careful to not repeatedly introduced the same superclass in our worklist). Here is what we do: For Givens: We add all their superclasses as Givens. For Wanteds: Generally speaking we want to be able to add superclasses of wanteds for two reasons: (1) Oportunities for improvement. Example: class (a ~ b) => C a b Wanted constraint is: C alpha beta We'd like to simply have C alpha alpha. Similar situations arise in relation to functional dependencies. (2) To have minimal constraints to quantify over: For instance, if our wanted constraint is (Eq a, Ord a) we'd only like to quantify over Ord a. To deal with (1) above we only add the superclasses of wanteds which may lead to improvement, that is: equality superclasses or superclasses with functional dependencies. We deal with (2) completely independently in TcSimplify. See Note [Minimize by SuperClasses] in TcSimplify. Moreover, in all cases the extra improvement constraints are Derived. Derived constraints have an identity (for now), but we don't do anything with their evidence. For instance they are never used to rewrite other constraints. See also [New Wanted Superclass Work] in TcInteract. For Deriveds: We do nothing. Here's an example that demonstrates why we chose to NOT add superclasses during simplification: [Comes from ticket #4497] class Num (RealOf t) => Normed t type family RealOf x Assume the generated wanted constraint is: RealOf e ~ e, Normed e If we were to be adding the superclasses during simplification we'd get: Num uf, Normed e, RealOf e ~ e, RealOf e ~ uf ==> e ~ uf, Num uf, Normed e, RealOf e ~ e ==> [Spontaneous solve] Num uf, Normed uf, RealOf uf ~ uf While looks exactly like our original constraint. If we add the superclass again we'd loop. By adding superclasses definitely only once, during canonicalisation, this situation can't happen. \begin{code}

    newSCWorkFromFlavored :: EvVar -> CtFlavor -> Class -> [Xi] -> TcS CanonicalCts
    -- Returns superclasses, see Note [Adding superclasses]
    newSCWorkFromFlavored ev orig_flavor cls xis
      | isDerived orig_flavor
      = return emptyCCan  -- Deriveds don't yield more superclasses because we will
                          -- add them transitively in the case of wanteds.

      | isGiven orig_flavor
      = do { let sc_theta = immSuperClasses cls xis
                 flavor   = orig_flavor
           ; sc_vars <- mapM newEvVar sc_theta
           ; _ <- zipWithM_ setEvBind sc_vars [EvSuperClass ev n | n <- [0..]]
           ; mkCanonicals flavor sc_vars }

      | isEmptyVarSet (tyVarsOfTypes xis)
      = return emptyCCan -- Wanteds with no variables yield no deriveds.
                         -- See Note [Improvement from Ground Wanteds]

      | otherwise -- Wanted case, just add those SC that can lead to improvement.
      = do { let sc_rec_theta = transSuperClasses cls xis
                 impr_theta   = filter is_improvement_pty sc_rec_theta
                 Wanted wloc  = orig_flavor
           ; der_ids <- mapM newDerivedId impr_theta
           ; mkCanonicals (Derived wloc) der_ids }

    is_improvement_pty :: PredType -> Bool
    -- Either it's an equality, or has some functional dependency
    is_improvement_pty (EqPred {})      = True
    is_improvement_pty (ClassP cls _ty) = not $ null fundeps
     where (_,fundeps,_,_,_,_) = classExtraBigSig cls
    is_improvement_pty _ = False

    canIP :: CtFlavor -> EvVar -> IPName Name -> TcType -> TcS CanonicalCts
    -- See Note [Canonical implicit parameter constraints] to see why we don't
    -- immediately canonicalize (flatten) IP constraints.
    canIP fl v nm ty
      = return $ singleCCan $ CIPCan { cc_id = v
                                     , cc_flavor = fl
                                     , cc_ip_nm = nm
                                     , cc_ip_ty = ty }

    -----------------
    canEq :: CtFlavor -> EvVar -> Type -> Type -> TcS CanonicalCts
    canEq fl cv ty1 ty2
      | tcEqType ty1 ty2	-- Dealing with equality here avoids
        	     	 	-- later spurious occurs checks for a~a
      = do { when (isWanted fl) (setCoBind cv ty1)
           ; return emptyCCan }

    -- If one side is a variable, orient and flatten,
    -- WITHOUT expanding type synonyms, so that we tend to
    -- substitute a ~ Age rather than a ~ Int when @type Age = Int@
    canEq fl cv ty1@(TyVarTy {}) ty2
      = do { untch <- getUntouchables
           ; canEqLeaf untch fl cv (classify ty1) (classify ty2) }
    canEq fl cv ty1 ty2@(TyVarTy {})
      = do { untch <- getUntouchables
           ; canEqLeaf untch fl cv (classify ty1) (classify ty2) }
          -- NB: don't use VarCls directly because tv1 or tv2 may be scolems!

    canEq fl cv (TyConApp fn tys) ty2
      | isSynFamilyTyCon fn, length tys == tyConArity fn
      = do { untch <- getUntouchables
           ; canEqLeaf untch fl cv (FunCls fn tys) (classify ty2) }
    canEq fl cv ty1 (TyConApp fn tys)
      | isSynFamilyTyCon fn, length tys == tyConArity fn
      = do { untch <- getUntouchables
           ; canEqLeaf untch fl cv (classify ty1) (FunCls fn tys) }

    canEq fl cv s1 s2
      | Just (t1a,t1b,t1c) <- splitCoPredTy_maybe s1,
        Just (t2a,t2b,t2c) <- splitCoPredTy_maybe s2
      = do { (v1,v2,v3)
                 <- if isWanted fl then                   -- Wanted
                        do { v1 <- newCoVar t1a t2a
                           ; v2 <- newCoVar t1b t2b
                           ; v3 <- newCoVar t1c t2c
                           ; let res_co = mkCoPredCo (mkCoVarCoercion v1)
                                            (mkCoVarCoercion v2) (mkCoVarCoercion v3)
                           ; setCoBind cv res_co
                           ; return (v1,v2,v3) }
                    else if isGiven fl then               -- Given
                             let co_orig = mkCoVarCoercion cv
                                 coa = mkCsel1Coercion co_orig
                                 cob = mkCsel2Coercion co_orig
                                 coc = mkCselRCoercion co_orig
                             in do { v1 <- newGivenCoVar t1a t2a coa
                                   ; v2 <- newGivenCoVar t1b t2b cob
                                   ; v3 <- newGivenCoVar t1c t2c coc
                                   ; return (v1,v2,v3) }
                    else                                  -- Derived
                        do { v1 <- newDerivedId (EqPred t1a t2a)
                           ; v2 <- newDerivedId (EqPred t1b t2b)
                           ; v3 <- newDerivedId (EqPred t1c t2c)
                           ; return (v1,v2,v3) }
           ; cc1 <- canEq fl v1 t1a t2a
           ; cc2 <- canEq fl v2 t1b t2b
           ; cc3 <- canEq fl v3 t1c t2c
           ; return (cc1 `andCCan` cc2 `andCCan` cc3) }

    -- Split up an equality between function types into two equalities.
    canEq fl cv (FunTy s1 t1) (FunTy s2 t2)
      = do { (argv, resv) <-
                 if isWanted fl then
                     do { argv <- newCoVar s1 s2
                        ; resv <- newCoVar t1 t2
                        ; setCoBind cv $
                          mkFunCoercion (mkCoVarCoercion argv) (mkCoVarCoercion resv)
                        ; return (argv,resv) }

                 else if isGiven fl then
                          let [arg,res] = decomposeCo 2 (mkCoVarCoercion cv)
                          in do { argv <- newGivenCoVar s1 s2 arg
                                ; resv <- newGivenCoVar t1 t2 res
                                ; return (argv,resv) }

                 else -- Derived
                     do { argv <- newDerivedId (EqPred s1 s2)
                        ; resv <- newDerivedId (EqPred t1 t2)
                        ; return (argv,resv) }

           ; cc1 <- canEq fl argv s1 s2 -- inherit original kinds and locations
           ; cc2 <- canEq fl resv t1 t2
           ; return (cc1 `andCCan` cc2) }

    canEq fl cv (PredTy (IParam n1 t1)) (PredTy (IParam n2 t2))
      | n1 == n2
      = if isWanted fl then
            do { v <- newCoVar t1 t2
               ; setCoBind cv $ mkIParamPredCo n1 (mkCoVarCoercion cv)
               ; canEq fl v t1 t2 }
        else return emptyCCan -- DV: How to decompose given IP coercions?

    canEq fl cv (PredTy (ClassP c1 tys1)) (PredTy (ClassP c2 tys2))
      | c1 == c2
      = if isWanted fl then
           do { vs <- zipWithM newCoVar tys1 tys2
              ; setCoBind cv $ mkClassPPredCo c1 (map mkCoVarCoercion vs)
              ; andCCans <$> zipWith3M (canEq fl) vs tys1 tys2
              }
        else return emptyCCan
      -- How to decompose given dictionary (and implicit parameter) coercions?
      -- You may think that the following is right:
      --    let cos = decomposeCo (length tys1) (mkCoVarCoercion cv)
      --    in  zipWith3M newGivOrDerCoVar tys1 tys2 cos
      -- But this assumes that the coercion is a type constructor-based
      -- coercion, and not a PredTy (ClassP cn cos) coercion. So we chose
      -- to not decompose these coercions. We have to get back to this
      -- when we clean up the Coercion API.

    canEq fl cv (TyConApp tc1 tys1) (TyConApp tc2 tys2)
      | isAlgTyCon tc1 && isAlgTyCon tc2
      , tc1 == tc2
      , length tys1 == length tys2
      = -- Generate equalities for each of the corresponding arguments
        do { argsv
                 <- if isWanted fl then
                        do { argsv <- zipWithM newCoVar tys1 tys2
                           ; setCoBind cv $
                             mkTyConCoercion tc1 (map mkCoVarCoercion argsv)
                           ; return argsv }

                    else if isGiven fl then
                        let cos = decomposeCo (length tys1) (mkCoVarCoercion cv)
                        in zipWith3M newGivenCoVar tys1 tys2 cos

                    else -- Derived
                        zipWithM (\t1 t2 -> newDerivedId (EqPred t1 t2)) tys1 tys2

           ; andCCans <$> zipWith3M (canEq fl) argsv tys1 tys2 }

    -- See Note [Equality between type applications]
    --     Note [Care with type applications] in TcUnify
    canEq fl cv ty1 ty2
      | Just (s1,t1) <- tcSplitAppTy_maybe ty1
      , Just (s2,t2) <- tcSplitAppTy_maybe ty2
        = do { (cv1,cv2) <-
                 if isWanted fl
                 then do { cv1 <- newCoVar s1 s2
                         ; cv2 <- newCoVar t1 t2
                         ; setCoBind cv $
                           mkAppCoercion (mkCoVarCoercion cv1) (mkCoVarCoercion cv2)
                         ; return (cv1,cv2) }

                 else if isGiven fl then
                        let co1 = mkLeftCoercion  $ mkCoVarCoercion cv
                            co2 = mkRightCoercion $ mkCoVarCoercion cv
                        in do { cv1 <- newGivenCoVar s1 s2 co1
                              ; cv2 <- newGivenCoVar t1 t2 co2
                              ; return (cv1,cv2) }
                 else -- Derived
                     do { cv1 <- newDerivedId (EqPred s1 s2)
                        ; cv2 <- newDerivedId (EqPred t1 t2)
                        ; return (cv1,cv2) }

             ; cc1 <- canEq fl cv1 s1 s2
             ; cc2 <- canEq fl cv2 t1 t2
             ; return (cc1 `andCCan` cc2) }

    canEq fl cv s1@(ForAllTy {}) s2@(ForAllTy {})
     | tcIsForAllTy s1, tcIsForAllTy s2,
       Wanted {} <- fl
     = canEqFailure fl cv
     | otherwise
     = do { traceTcS "Ommitting decomposition of given polytype equality" (pprEq s1 s2)
          ; return emptyCCan }

    -- Finally expand any type synonym applications.
    canEq fl cv ty1 ty2 | Just ty1' <- tcView ty1 = canEq fl cv ty1' ty2
    canEq fl cv ty1 ty2 | Just ty2' <- tcView ty2 = canEq fl cv ty1 ty2'
    canEq fl cv _ _                               = canEqFailure fl cv

    canEqFailure :: CtFlavor -> EvVar -> TcS CanonicalCts
    canEqFailure fl cv = return (singleCCan (mkFrozenError fl cv))

\end{code} Note [Equality between type applications] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If we see an equality of the form s1 t1 ~ s2 t2 we can always split it up into s1 ~ s2 /\ t1 ~ t2, since s1 and s2 can't be type functions (type functions use the TyConApp constructor, which never shows up as the LHS of an AppTy). Other than type functions, types in Haskell are always (1) generative: a b ~ c d implies a ~ c, since different type constructors always generate distinct types (2) injective: a b ~ a d implies b ~ d; we never generate the same type from different type arguments. Note [Canonical ordering for equality constraints] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Implemented as (<+=) below: - Type function applications always come before anything else. - Variables always come before non-variables (other than type function applications). Note that we don't need to unfold type synonyms on the RHS to check the ordering; that is, in the rules above it's OK to consider only whether something is *syntactically* a type function application or not. To illustrate why this is OK, suppose we have an equality of the form 'tv ~ S a b c', where S is a type synonym which expands to a top-level application of the type function F, something like type S a b c = F d e Then to canonicalize 'tv ~ S a b c' we flatten the RHS, and since S's expansion contains type function applications the flattener will do the expansion and then generate a skolem variable for the type function application, so we end up with something like this: tv ~ x F d e ~ x where x is the skolem variable. This is one extra equation than absolutely necessary (we could have gotten away with just 'F d e ~ tv' if we had noticed that S expanded to a top-level type function application and flipped it around in the first place) but this way keeps the code simpler. Unlike the OutsideIn(X) draft of May 7, 2010, we do not care about the ordering of tv ~ tv constraints. There are several reasons why we might: (1) In order to be able to extract a substitution that doesn't mention untouchable variables after we are done solving, we might prefer to put touchable variables on the left. However, in and of itself this isn't necessary; we can always re-orient equality constraints at the end if necessary when extracting a substitution. (2) To ensure termination we might think it necessary to put variables in lexicographic order. However, this isn't actually necessary as outlined below. While building up an inert set of canonical constraints, we maintain the invariant that the equality constraints in the inert set form an acyclic rewrite system when viewed as L-R rewrite rules. Moreover, the given constraints form an idempotent substitution (i.e. none of the variables on the LHS occur in any of the RHS's, and type functions never show up in the RHS at all), the wanted constraints also form an idempotent substitution, and finally the LHS of a given constraint never shows up on the RHS of a wanted constraint. There may, however, be a wanted LHS that shows up in a given RHS, since we do not rewrite given constraints with wanted constraints. Suppose we have an inert constraint set tg_1 ~ xig_1 -- givens tg_2 ~ xig_2 ... tw_1 ~ xiw_1 -- wanteds tw_2 ~ xiw_2 ... where each t_i can be either a type variable or a type function application. Now suppose we take a new canonical equality constraint, t' ~ xi' (note among other things this means t' does not occur in xi') and try to react it with the existing inert set. We show by induction on the number of t_i which occur in t' ~ xi' that this process will terminate. There are several ways t' ~ xi' could react with an existing constraint: TODO: finish this proof. The below was for the case where the entire inert set is an idempotent subustitution... (b) We could have t' = t_j for some j. Then we obtain the new equality xi_j ~ xi'; note that neither xi_j or xi' contain t_j. We now canonicalize the new equality, which may involve decomposing it into several canonical equalities, and recurse on these. However, none of the new equalities will contain t_j, so they have fewer occurrences of the t_i than the original equation. (a) We could have t_j occurring in xi' for some j, with t' /= t_j. Then we substitute xi_j for t_j in xi' and continue. However, since none of the t_i occur in xi_j, we have decreased the number of t_i that occur in xi', since we eliminated t_j and did not introduce any new ones. \begin{code}

    data TypeClassifier
      = FskCls TcTyVar      -- ^ Flatten skolem
      | VarCls TcTyVar      -- ^ Non-flatten-skolem variable
      | FunCls TyCon [Type] -- ^ Type function, exactly saturated
      | OtherCls TcType     -- ^ Neither of the above

    unClassify :: TypeClassifier -> TcType
    unClassify (VarCls tv)      = TyVarTy tv
    unClassify (FskCls tv) = TyVarTy tv
    unClassify (FunCls fn tys)  = TyConApp fn tys
    unClassify (OtherCls ty)    = ty

    classify :: TcType -> TypeClassifier

    classify (TyVarTy tv)
      | isTcTyVar tv,
        FlatSkol {} <- tcTyVarDetails tv = FskCls tv
      | otherwise                        = VarCls tv
    classify (TyConApp tc tys) | isSynFamilyTyCon tc
                               , tyConArity tc == length tys
                               = FunCls tc tys
    classify ty                | Just ty' <- tcView ty
    	                   = case classify ty' of
                                   OtherCls {} -> OtherCls ty
                                   var_or_fn   -> var_or_fn
                               | otherwise
                               = OtherCls ty

    -- See note [Canonical ordering for equality constraints].
    reOrient :: CtFlavor -> TypeClassifier -> TypeClassifier -> Bool
    -- (t1 `reOrient` t2) responds True
    --   iff we should flip to (t2~t1)
    -- We try to say False if possible, to minimise evidence generation
    --
    -- Postcondition: After re-orienting, first arg is not OTherCls
    reOrient _fl (OtherCls {}) (FunCls {})   = True
    reOrient _fl (OtherCls {}) (FskCls {})   = True
    reOrient _fl (OtherCls {}) (VarCls {})   = True
    reOrient _fl (OtherCls {}) (OtherCls {}) = panic "reOrient"  -- One must be Var/Fun

    reOrient _fl (FunCls {})   (VarCls _tv)  = False
      -- But consider the following variation: isGiven fl && isMetaTyVar tv

      -- See Note [No touchables as FunEq RHS] in TcSMonad
    reOrient _fl (FunCls {}) _                = False             -- Fun/Other on rhs

    reOrient _fl (VarCls {}) (FunCls {})      = True

    reOrient _fl (VarCls {}) (FskCls {})      = False

    reOrient _fl (VarCls {})  (OtherCls {})   = False
    reOrient _fl (VarCls tv1)  (VarCls tv2)
      | isMetaTyVar tv2 && not (isMetaTyVar tv1) = True
      | otherwise                                = False
      -- Just for efficiency, see CTyEqCan invariants

    reOrient _fl (FskCls {}) (VarCls tv2)     = isMetaTyVar tv2
      -- Just for efficiency, see CTyEqCan invariants

    reOrient _fl (FskCls {}) (FskCls {})     = False
    reOrient _fl (FskCls {}) (FunCls {})     = True
    reOrient _fl (FskCls {}) (OtherCls {})   = False

    ------------------
    canEqLeaf :: TcsUntouchables
              -> CtFlavor -> CoVar
              -> TypeClassifier -> TypeClassifier -> TcS CanonicalCts
    -- Canonicalizing "leaf" equality constraints which cannot be
    -- decomposed further (ie one of the types is a variable or
    -- saturated type function application).

      -- Preconditions:
      --    * one of the two arguments is not OtherCls
      --    * the two types are not equal (looking through synonyms)
    canEqLeaf _untch fl cv cls1 cls2
      | cls1 `re_orient` cls2
      = do { cv' <- if isWanted fl
                    then do { cv' <- newCoVar s2 s1
                            ; setCoBind cv $ mkSymCoercion (mkCoVarCoercion cv')
                            ; return cv' }
                    else if isGiven fl then
                             newGivenCoVar s2 s1 (mkSymCoercion (mkCoVarCoercion cv))
                    else -- Derived
                        newDerivedId (EqPred s2 s1)
           ; canEqLeafOriented fl cv' cls2 s1 }

      | otherwise
      = do { traceTcS "canEqLeaf" (ppr (unClassify cls1) $$ ppr (unClassify cls2))
           ; canEqLeafOriented fl cv cls1 s2 }
      where
        re_orient = reOrient fl
        s1 = unClassify cls1
        s2 = unClassify cls2

    ------------------
    canEqLeafOriented :: CtFlavor -> CoVar
                      -> TypeClassifier -> TcType -> TcS CanonicalCts
    -- First argument is not OtherCls
    canEqLeafOriented fl cv cls1@(FunCls fn tys1) s2         -- cv : F tys1
      | let k1 = kindAppResult (tyConKind fn) tys1,
        let k2 = typeKind s2,
        not (k1 `compatKind` k2) -- Establish the kind invariant for CFunEqCan
      = canEqFailure fl cv
        -- Eagerly fails, see Note [Kind errors] in TcInteract

      | otherwise
      = ASSERT2( isSynFamilyTyCon fn, ppr (unClassify cls1) )
        do { (xis1,cos1,ccs1) <- flattenMany fl tys1 -- Flatten type function arguments
                                                     -- cos1 :: xis1 ~ tys1
           ; (xi2, co2, ccs2) <- flatten fl s2       -- Flatten entire RHS
                                                     -- co2  :: xi2 ~ s2
           ; let ccs = ccs1 `andCCan` ccs2
                 no_flattening_happened = isEmptyCCan ccs
           ; cv_new <- if no_flattening_happened then return cv
                       else if isGiven fl        then return cv
                       else if isWanted fl then
                             do { cv' <- newCoVar (unClassify (FunCls fn xis1)) xi2
                                     -- cv' : F xis ~ xi2
                                ; let -- fun_co :: F xis1 ~ F tys1
                                     fun_co = mkTyConCoercion fn cos1
                                     -- want_co :: F tys1 ~ s2
                                     want_co = mkSymCoercion fun_co
                                               `mkTransCoercion` mkCoVarCoercion cv'
                                               `mkTransCoercion` co2
                                ; setCoBind cv  want_co
                                ; return cv' }
                       else -- Derived
                           newDerivedId (EqPred (unClassify (FunCls fn xis1)) xi2)

           ; let final_cc = CFunEqCan { cc_id     = cv_new
                                      , cc_flavor = fl
                                      , cc_fun    = fn
                                      , cc_tyargs = xis1
                                      , cc_rhs    = xi2 }
           ; return $ ccs `extendCCans` final_cc }

    -- Otherwise, we have a variable on the left, so call canEqLeafTyVarLeft
    canEqLeafOriented fl cv (FskCls tv) s2
      = canEqLeafTyVarLeft fl cv tv s2
    canEqLeafOriented fl cv (VarCls tv) s2
      = canEqLeafTyVarLeft fl cv tv s2
    canEqLeafOriented _ cv (OtherCls ty1) ty2
      = pprPanic "canEqLeaf" (ppr cv $$ ppr ty1 $$ ppr ty2)

    canEqLeafTyVarLeft :: CtFlavor -> CoVar -> TcTyVar -> TcType -> TcS CanonicalCts
    -- Establish invariants of CTyEqCans
    canEqLeafTyVarLeft fl cv tv s2       -- cv : tv ~ s2
      | not (k1 `compatKind` k2) -- Establish the kind invariant for CTyEqCan
      = canEqFailure fl cv
           -- Eagerly fails, see Note [Kind errors] in TcInteract
      | otherwise
      = do { (xi2, co, ccs2) <- flatten fl s2  -- Flatten RHS   co : xi2 ~ s2
           ; mxi2' <- canOccursCheck fl tv xi2 -- Do an occurs check, and return a possibly
                                               -- unfolded version of the RHS, if we had to
                                               -- unfold any type synonyms to get rid of tv.
           ; case mxi2' of {
               Nothing   -> canEqFailure fl cv ;
               Just xi2' ->
        do { let no_flattening_happened = isEmptyCCan ccs2
           ; cv_new <- if no_flattening_happened then return cv
                       else if isGiven fl        then return cv
                       else if isWanted fl then
                             do { cv' <- newCoVar (mkTyVarTy tv) xi2'  -- cv' : tv ~ xi2
                                ; setCoBind cv  (mkCoVarCoercion cv' `mkTransCoercion` co)
                                ; return cv' }
                       else -- Derived
                           newDerivedId (EqPred (mkTyVarTy tv) xi2')

           ; return $ ccs2 `extendCCans` CTyEqCan { cc_id     = cv_new
                                                  , cc_flavor = fl
                                                  , cc_tyvar  = tv
                                                  , cc_rhs    = xi2' } } } }
      where
        k1 = tyVarKind tv
        k2 = typeKind s2

    -- See Note [Type synonyms and canonicalization].
    -- Check whether the given variable occurs in the given type.  We may
    -- have needed to do some type synonym unfolding in order to get rid
    -- of the variable, so we also return the unfolded version of the
    -- type, which is guaranteed to be syntactically free of the given
    -- type variable.  If the type is already syntactically free of the
    -- variable, then the same type is returned.
    --
    -- Precondition: the two types are not equal (looking though synonyms)
    canOccursCheck :: CtFlavor -> TcTyVar -> Xi -> TcS (Maybe Xi)
    canOccursCheck _gw tv xi = return (expandAway tv xi)

\end{code} @expandAway tv xi@ expands synonyms in xi just enough to get rid of occurrences of tv, if that is possible; otherwise, it returns Nothing. For example, suppose we have type F a b = [a] Then expandAway b (F Int b) = Just [Int] but expandAway a (F a Int) = Nothing We don't promise to do the absolute minimum amount of expanding necessary, but we try not to do expansions we don't need to. We prefer doing inner expansions first. For example, type F a b = (a, Int, a, [a]) type G b = Char We have expandAway b (F (G b)) = F Char even though we could also expand F to get rid of b. \begin{code}

    expandAway :: TcTyVar -> Xi -> Maybe Xi
    expandAway tv t@(TyVarTy tv')
      | tv == tv' = Nothing
      | otherwise = Just t
    expandAway tv xi
      | not (tv `elemVarSet` tyVarsOfType xi) = Just xi
    expandAway tv (AppTy ty1 ty2)
      = do { ty1' <- expandAway tv ty1
           ; ty2' <- expandAway tv ty2
           ; return (mkAppTy ty1' ty2') }
    -- mkAppTy <$> expandAway tv ty1 <*> expandAway tv ty2
    expandAway tv (FunTy ty1 ty2)
      = do { ty1' <- expandAway tv ty1
           ; ty2' <- expandAway tv ty2
           ; return (mkFunTy ty1' ty2') }
    -- mkFunTy <$> expandAway tv ty1 <*> expandAway tv ty2
    expandAway tv ty@(ForAllTy {})
      = let (tvs,rho) = splitForAllTys ty
            tvs_knds  = map tyVarKind tvs
        in if tv `elemVarSet` tyVarsOfTypes tvs_knds then
           -- Can't expand away the kinds unless we create
           -- fresh variables which we don't want to do at this point.
               Nothing
           else do { rho' <- expandAway tv rho
                   ; return (mkForAllTys tvs rho') }
    expandAway tv (PredTy pred)
      = do { pred' <- expandAwayPred tv pred
           ; return (PredTy pred') }
    -- For a type constructor application, first try expanding away the
    -- offending variable from the arguments.  If that doesn't work, next
    -- see if the type constructor is a type synonym, and if so, expand
    -- it and try again.
    expandAway tv ty@(TyConApp tc tys)
      = (mkTyConApp tc <$> mapM (expandAway tv) tys) <|> (tcView ty >>= expandAway tv)

    expandAwayPred :: TcTyVar -> TcPredType -> Maybe TcPredType
    expandAwayPred tv (ClassP cls tys)
      = do { tys' <- mapM (expandAway tv) tys; return (ClassP cls tys') }
    expandAwayPred tv (EqPred ty1 ty2)
      = do { ty1' <- expandAway tv ty1
           ; ty2' <- expandAway tv ty2
           ; return (EqPred ty1' ty2') }
    expandAwayPred tv (IParam nm ty)
      = do { ty' <- expandAway tv ty
           ; return (IParam nm ty') }

\end{code} Note [Type synonyms and canonicalization] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We treat type synonym applications as xi types, that is, they do not count as type function applications. However, we do need to be a bit careful with type synonyms: like type functions they may not be generative or injective. However, unlike type functions, they are parametric, so there is no problem in expanding them whenever we see them, since we do not need to know anything about their arguments in order to expand them; this is what justifies not having to treat them as specially as type function applications. The thing that causes some subtleties is that we prefer to leave type synonym applications *unexpanded* whenever possible, in order to generate better error messages. If we encounter an equality constraint with type synonym applications on both sides, or a type synonym application on one side and some sort of type application on the other, we simply must expand out the type synonyms in order to continue decomposing the equality constraint into primitive equality constraints. For example, suppose we have type F a = [Int] and we encounter the equality F a ~ [b] In order to continue we must expand F a into [Int], giving us the equality [Int] ~ [b] which we can then decompose into the more primitive equality constraint Int ~ b. However, if we encounter an equality constraint with a type synonym application on one side and a variable on the other side, we should NOT (necessarily) expand the type synonym, since for the purpose of good error messages we want to leave type synonyms unexpanded as much as possible. However, there is a subtle point with type synonyms and the occurs check that takes place for equality constraints of the form tv ~ xi. As an example, suppose we have type F a = Int and we come across the equality constraint a ~ F a This should not actually fail the occurs check, since expanding out the type synonym results in the legitimate equality constraint a ~ Int. We must actually do this expansion, because unifying a with F a will lead the type checker into infinite loops later. Put another way, canonical equality constraints should never *syntactically* contain the LHS variable in the RHS type. However, we don't always need to expand type synonyms when doing an occurs check; for example, the constraint a ~ F b is obviously fine no matter what F expands to. And in this case we would rather unify a with F b (rather than F b's expansion) in order to get better error messages later. So, when doing an occurs check with a type synonym application on the RHS, we use some heuristics to find an expansion of the RHS which does not contain the variable from the LHS. In particular, given a ~ F t1 ... tn we first try expanding each of the ti to types which no longer contain a. If this turns out to be impossible, we next try expanding F itself, and so on. %************************************************************************ %* * %* Functional dependencies, instantiation of equations %* * %************************************************************************ \begin{code}

    rewriteWithFunDeps :: [Equation]
                       -> [Xi] -> CtFlavor
                       -> TcS (Maybe ([Xi], [Coercion], CanonicalCts))
    rewriteWithFunDeps eqn_pred_locs xis fl
     = do { fd_ev_poss <- mapM (instFunDepEqn fl) eqn_pred_locs
          ; let fd_ev_pos :: [(Int,FlavoredEvVar)]
                fd_ev_pos = concat fd_ev_poss
                (rewritten_xis, cos) = unzip (rewriteDictParams fd_ev_pos xis)
          ; fds <- mapM (\(_,fev) -> mkCanonicalFEV fev) fd_ev_pos
          ; let fd_work = unionManyBags fds
          ; if isEmptyBag fd_work
            then return Nothing
            else return (Just (rewritten_xis, cos, fd_work)) }

    instFunDepEqn :: CtFlavor -- Precondition: Only Wanted or Derived
                  -> Equation
                  -> TcS [(Int, FlavoredEvVar)]
    -- Post: Returns the position index as well as the corresponding FunDep equality
    instFunDepEqn fl (FDEqn { fd_qtvs = qtvs, fd_eqs = eqs
                            , fd_pred1 = d1, fd_pred2 = d2 })
      = do { let tvs = varSetElems qtvs
           ; tvs' <- mapM instFlexiTcS tvs
           ; let subst = zipTopTvSubst tvs (mkTyVarTys tvs')
           ; mapM (do_one subst) eqs }
      where
        fl' = case fl of
                 Given _     -> panic "mkFunDepEqns"
                 Wanted  loc -> Wanted  (push_ctx loc)
                 Derived loc -> Derived (push_ctx loc)

        push_ctx loc = pushErrCtxt FunDepOrigin (False, mkEqnMsg d1 d2) loc

        do_one subst (FDEq { fd_pos = i, fd_ty_left = ty1, fd_ty_right = ty2 })
           = do { let sty1 = substTy subst ty1
                      sty2 = substTy subst ty2
                ; ev <- newCoVar sty1 sty2
                ; return (i, mkEvVarX ev fl') }

    rewriteDictParams :: [(Int,FlavoredEvVar)] -- A set of coercions : (pos, ty' ~ ty)
                      -> [Type]                -- A sequence of types: tys
                      -> [(Type,Coercion)]     -- Returns            : [(ty', co : ty' ~ ty)]
    rewriteDictParams param_eqs tys
      = zipWith do_one tys [0..]
      where
        do_one :: Type -> Int -> (Type,Coercion)
        do_one ty n = case lookup n param_eqs of
                        Just wev -> (get_fst_ty wev, mkCoVarCoercion (evVarOf wev))
                        Nothing  -> (ty,ty)		-- Identity

        get_fst_ty wev = case evVarOfPred wev of
                              EqPred ty1 _ -> ty1
                              _ -> panic "rewriteDictParams: non equality fundep"

    mkEqnMsg :: (TcPredType, SDoc) -> (TcPredType, SDoc) -> TidyEnv
             -> TcM (TidyEnv, SDoc)
    mkEqnMsg (pred1,from1) (pred2,from2) tidy_env
      = do  { zpred1 <- TcM.zonkTcPredType pred1
            ; zpred2 <- TcM.zonkTcPredType pred2
    	; let { tpred1 = tidyPred tidy_env zpred1
                  ; tpred2 = tidyPred tidy_env zpred2 }
    	; let msg = vcat [ptext (sLit "When using functional dependencies to combine"),
    			  nest 2 (sep [ppr tpred1 <> comma, nest 2 from1]),
    			  nest 2 (sep [ppr tpred2 <> comma, nest 2 from2])]
    	; return (tidy_env, msg) }

\end{code}