typecheck/TcBinds.hs

    {-
    (c) The University of Glasgow 2006
    (c) The GRASP/AQUA Project, Glasgow University, 1992-1998

    \section[TcBinds]{TcBinds}
    -}

    {-# LANGUAGE CPP, RankNTypes, ScopedTypeVariables #-}

    module TcBinds ( tcLocalBinds, tcTopBinds, tcRecSelBinds,
                     tcHsBootSigs, tcPolyCheck,
                     PragFun, tcSpecPrags, tcVectDecls, mkPragFun,
                     TcSigInfo(..), TcSigFun,
                     instTcTySig, instTcTySigFromId, findScopedTyVars,
                     badBootDeclErr, mkExport ) where

    import {-# SOURCE #-} TcMatches ( tcGRHSsPat, tcMatchesFun )
    import {-# SOURCE #-} TcExpr  ( tcMonoExpr )
    import {-# SOURCE #-} TcPatSyn ( tcInferPatSynDecl, tcCheckPatSynDecl, tcPatSynBuilderBind )
    import DynFlags
    import HsSyn
    import HscTypes( isHsBootOrSig )
    import TcRnMonad
    import TcEnv
    import TcUnify
    import TcSimplify
    import TcEvidence
    import TcHsType
    import TcPat
    import TcMType
    import ConLike
    import FamInstEnv( normaliseType )
    import FamInst( tcGetFamInstEnvs )
    import Type( pprSigmaTypeExtraCts )
    import TyCon
    import TcType
    import TysPrim
    import Id
    import Var
    import VarSet
    import VarEnv( TidyEnv )
    import Module
    import Name
    import NameSet
    import NameEnv
    import SrcLoc
    import Bag
    import ListSetOps
    import ErrUtils
    import Digraph
    import Maybes
    import Util
    import BasicTypes
    import Outputable
    import FastString
    import Type(mkStrLitTy)
    import PrelNames(ipClassName)
    import TcValidity (checkValidType)

    import Control.Monad
    import Data.List (partition)

    #include "HsVersions.h"

    {-
    ************************************************************************
    *                                                                      *
    \subsection{Type-checking bindings}
    *                                                                      *
    ************************************************************************

    @tcBindsAndThen@ typechecks a @HsBinds@.  The "and then" part is because
    it needs to know something about the {\em usage} of the things bound,
    so that it can create specialisations of them.  So @tcBindsAndThen@
    takes a function which, given an extended environment, E, typechecks
    the scope of the bindings returning a typechecked thing and (most
    important) an LIE.  It is this LIE which is then used as the basis for
    specialising the things bound.

    @tcBindsAndThen@ also takes a "combiner" which glues together the
    bindings and the "thing" to make a new "thing".

    The real work is done by @tcBindWithSigsAndThen@.

    Recursive and non-recursive binds are handled in essentially the same
    way: because of uniques there are no scoping issues left.  The only
    difference is that non-recursive bindings can bind primitive values.

    Even for non-recursive binding groups we add typings for each binder
    to the LVE for the following reason.  When each individual binding is
    checked the type of its LHS is unified with that of its RHS; and
    type-checking the LHS of course requires that the binder is in scope.

    At the top-level the LIE is sure to contain nothing but constant
    dictionaries, which we resolve at the module level.

    Note [Polymorphic recursion]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The game plan for polymorphic recursion in the code above is

            * Bind any variable for which we have a type signature
              to an Id with a polymorphic type.  Then when type-checking
              the RHSs we'll make a full polymorphic call.

    This fine, but if you aren't a bit careful you end up with a horrendous
    amount of partial application and (worse) a huge space leak. For example:

            f :: Eq a => [a] -> [a]
            f xs = ...f...

    If we don't take care, after typechecking we get

            f = /\a -> \d::Eq a -> let f' = f a d
                                   in
                                   \ys:[a] -> ...f'...

    Notice the the stupid construction of (f a d), which is of course
    identical to the function we're executing.  In this case, the
    polymorphic recursion isn't being used (but that's a very common case).
    This can lead to a massive space leak, from the following top-level defn
    (post-typechecking)

            ff :: [Int] -> [Int]
            ff = f Int dEqInt

    Now (f dEqInt) evaluates to a lambda that has f' as a free variable; but
    f' is another thunk which evaluates to the same thing... and you end
    up with a chain of identical values all hung onto by the CAF ff.

            ff = f Int dEqInt

               = let f' = f Int dEqInt in \ys. ...f'...

               = let f' = let f' = f Int dEqInt in \ys. ...f'...
                          in \ys. ...f'...

    Etc.

    NOTE: a bit of arity anaysis would push the (f a d) inside the (\ys...),
    which would make the space leak go away in this case

    Solution: when typechecking the RHSs we always have in hand the
    *monomorphic* Ids for each binding.  So we just need to make sure that
    if (Method f a d) shows up in the constraints emerging from (...f...)
    we just use the monomorphic Id.  We achieve this by adding monomorphic Ids
    to the "givens" when simplifying constraints.  That's what the "lies_avail"
    is doing.

    Then we get

            f = /\a -> \d::Eq a -> letrec
                                     fm = \ys:[a] -> ...fm...
                                   in
                                   fm
    -}

    tcTopBinds :: HsValBinds Name -> TcM (TcGblEnv, TcLclEnv)
    -- The TcGblEnv contains the new tcg_binds and tcg_spects
    -- The TcLclEnv has an extended type envt for the new bindings
    tcTopBinds (ValBindsOut binds sigs)
      = do  { -- Pattern synonym bindings populate the global environment
              (binds', (tcg_env, tcl_env)) <- tcValBinds TopLevel binds sigs $
                do { gbl <- getGblEnv
                   ; lcl <- getLclEnv
                   ; return (gbl, lcl) }
            ; specs <- tcImpPrags sigs   -- SPECIALISE prags for imported Ids

            ; let { tcg_env' = tcg_env { tcg_binds = foldr (unionBags . snd)
                                                           (tcg_binds tcg_env)
                                                           binds'
                                       , tcg_imp_specs = specs ++ tcg_imp_specs tcg_env } }

            ; return (tcg_env', tcl_env) }
            -- The top level bindings are flattened into a giant
            -- implicitly-mutually-recursive LHsBinds

    tcTopBinds (ValBindsIn {}) = panic "tcTopBinds"

    tcRecSelBinds :: HsValBinds Name -> TcM TcGblEnv
    tcRecSelBinds (ValBindsOut binds sigs)
      = tcExtendGlobalValEnv [sel_id | L _ (IdSig sel_id) <- sigs] $
        do { (rec_sel_binds, tcg_env) <- discardWarnings (tcValBinds TopLevel binds sigs getGblEnv)
           ; let tcg_env'
                  | isHsBootOrSig (tcg_src tcg_env) = tcg_env
                  | otherwise = tcg_env { tcg_binds = foldr (unionBags . snd)
                                                            (tcg_binds tcg_env)
                                                            rec_sel_binds }
                  -- Do not add the code for record-selector bindings when
                  -- compiling hs-boot files
           ; return tcg_env' }
    tcRecSelBinds (ValBindsIn {}) = panic "tcRecSelBinds"

    tcHsBootSigs :: HsValBinds Name -> TcM [Id]
    -- A hs-boot file has only one BindGroup, and it only has type
    -- signatures in it.  The renamer checked all this
    tcHsBootSigs (ValBindsOut binds sigs)
      = do  { checkTc (null binds) badBootDeclErr
            ; concat <$> mapM (addLocM tc_boot_sig) (filter isTypeLSig sigs) }
      where
        tc_boot_sig (TypeSig lnames ty _) = mapM f lnames
          where
            f (L _ name) = do  { sigma_ty <- tcHsSigType (FunSigCtxt name) ty
                               ; return (mkVanillaGlobal name sigma_ty) }
            -- Notice that we make GlobalIds, not LocalIds
        tc_boot_sig s = pprPanic "tcHsBootSigs/tc_boot_sig" (ppr s)
    tcHsBootSigs groups = pprPanic "tcHsBootSigs" (ppr groups)

    badBootDeclErr :: MsgDoc
    badBootDeclErr = ptext (sLit "Illegal declarations in an hs-boot file")

    ------------------------
    tcLocalBinds :: HsLocalBinds Name -> TcM thing
                 -> TcM (HsLocalBinds TcId, thing)

    tcLocalBinds EmptyLocalBinds thing_inside
      = do  { thing <- thing_inside
            ; return (EmptyLocalBinds, thing) }

    tcLocalBinds (HsValBinds (ValBindsOut binds sigs)) thing_inside
      = do  { (binds', thing) <- tcValBinds NotTopLevel binds sigs thing_inside
            ; return (HsValBinds (ValBindsOut binds' sigs), thing) }
    tcLocalBinds (HsValBinds (ValBindsIn {})) _ = panic "tcLocalBinds"

    tcLocalBinds (HsIPBinds (IPBinds ip_binds _)) thing_inside
      = do  { ipClass <- tcLookupClass ipClassName
            ; (given_ips, ip_binds') <-
                mapAndUnzipM (wrapLocSndM (tc_ip_bind ipClass)) ip_binds

            -- If the binding binds ?x = E, we  must now
            -- discharge any ?x constraints in expr_lie
            -- See Note [Implicit parameter untouchables]
            ; (ev_binds, result) <- checkConstraints (IPSkol ips)
                                      [] given_ips thing_inside

            ; return (HsIPBinds (IPBinds ip_binds' ev_binds), result) }
      where
        ips = [ip | L _ (IPBind (Left (L _ ip)) _) <- ip_binds]

            -- I wonder if we should do these one at at time
            -- Consider     ?x = 4
            --              ?y = ?x + 1
        tc_ip_bind ipClass (IPBind (Left (L _ ip)) expr)
           = do { ty <- newFlexiTyVarTy openTypeKind
                ; let p = mkStrLitTy $ hsIPNameFS ip
                ; ip_id <- newDict ipClass [ p, ty ]
                ; expr' <- tcMonoExpr expr ty
                ; let d = toDict ipClass p ty `fmap` expr'
                ; return (ip_id, (IPBind (Right ip_id) d)) }
        tc_ip_bind _ (IPBind (Right {}) _) = panic "tc_ip_bind"

        -- Coerces a `t` into a dictionry for `IP "x" t`.
        -- co : t -> IP "x" t
        toDict ipClass x ty = HsWrap $ mkWpCast $ TcCoercion $
                              wrapIP $ mkClassPred ipClass [x,ty]

    {-
    Note [Implicit parameter untouchables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We add the type variables in the types of the implicit parameters
    as untouchables, not so much because we really must not unify them,
    but rather because we otherwise end up with constraints like this
        Num alpha, Implic { wanted = alpha ~ Int }
    The constraint solver solves alpha~Int by unification, but then
    doesn't float that solved constraint out (it's not an unsolved
    wanted).  Result disaster: the (Num alpha) is again solved, this
    time by defaulting.  No no no.

    However [Oct 10] this is all handled automatically by the
    untouchable-range idea.

    Note [Placeholder PatSyn kinds]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider this (Trac #9161)

      {-# LANGUAGE PatternSynonyms, DataKinds #-}
      pattern A = ()
      b :: A
      b = undefined

    Here, the type signature for b mentions A.  But A is a pattern
    synonym, which is typechecked (for very good reasons; a view pattern
    in the RHS may mention a value binding) as part of a group of
    bindings.  It is entirely resonable to reject this, but to do so
    we need A to be in the kind environment when kind-checking the signature for B.

    Hence the tcExtendKindEnv2 patsyn_placeholder_kinds, which adds a binding
        A -> AGlobal (AConLike (PatSynCon _|_))
    to the environment. Then TcHsType.tcTyVar will find A in the kind environment,
    and will give a 'wrongThingErr' as a result.  But the lookup of A won't fail.

    The _|_ (= panic "fakePatSynCon") works because the wrongThingErr call, in
    tcTyVar, doesn't look inside the TcTyThing.
    -}

    tcValBinds :: TopLevelFlag
               -> [(RecFlag, LHsBinds Name)] -> [LSig Name]
               -> TcM thing
               -> TcM ([(RecFlag, LHsBinds TcId)], thing)

    tcValBinds top_lvl binds sigs thing_inside
      = do  {  -- Typecheck the signature
            ; (poly_ids, sig_fn, nwc_tvs) <- tcExtendKindEnv2 patsyn_placeholder_kinds $
                                             -- See Note [Placeholder PatSyn kinds]
                                             tcTySigs sigs

            ; let prag_fn = mkPragFun sigs (foldr (unionBags . snd) emptyBag binds)

                    -- Extend the envt right away with all
                    -- the Ids declared with type signatures
                    -- Use tcExtendIdEnv3 to avoid extending the TcIdBinder stack
            ; tcExtendIdEnv3 [(idName id, id) | id <- poly_ids] (mkVarSet nwc_tvs) $ do
                { (binds', (extra_binds', thing)) <- tcBindGroups top_lvl sig_fn prag_fn binds $ do
                       { thing <- thing_inside
                         -- See Note [Pattern synonym wrappers don't yield dependencies]
                       ; patsyn_workers <- mapM tcPatSynBuilderBind patsyns
                       ; let extra_binds = [ (NonRecursive, worker) | worker <- patsyn_workers ]
                       ; return (extra_binds, thing) }
                 ; return (binds' ++ extra_binds', thing) }}
      where
        patsyns
          = [psb | (_, lbinds) <- binds, L _ (PatSynBind psb) <- bagToList lbinds]
        patsyn_placeholder_kinds -- See Note [Placeholder PatSyn kinds]
          = [(name, placeholder_patsyn_tything)| PSB{ psb_id = L _ name } <- patsyns ]
        placeholder_patsyn_tything
          = AGlobal $ AConLike $ PatSynCon $ panic "fakePatSynCon"

    ------------------------
    tcBindGroups :: TopLevelFlag -> TcSigFun -> PragFun
                 -> [(RecFlag, LHsBinds Name)] -> TcM thing
                 -> TcM ([(RecFlag, LHsBinds TcId)], thing)
    -- Typecheck a whole lot of value bindings,
    -- one strongly-connected component at a time
    -- Here a "strongly connected component" has the strightforward
    -- meaning of a group of bindings that mention each other,
    -- ignoring type signatures (that part comes later)

    tcBindGroups _ _ _ [] thing_inside
      = do  { thing <- thing_inside
            ; return ([], thing) }

    tcBindGroups top_lvl sig_fn prag_fn (group : groups) thing_inside
      = do  { (group', (groups', thing))
                    <- tc_group top_lvl sig_fn prag_fn group $
                       tcBindGroups top_lvl sig_fn prag_fn groups thing_inside
            ; return (group' ++ groups', thing) }

    ------------------------
    tc_group :: forall thing.
                TopLevelFlag -> TcSigFun -> PragFun
             -> (RecFlag, LHsBinds Name) -> TcM thing
             -> TcM ([(RecFlag, LHsBinds TcId)], thing)

    -- Typecheck one strongly-connected component of the original program.
    -- We get a list of groups back, because there may
    -- be specialisations etc as well

    tc_group top_lvl sig_fn prag_fn (NonRecursive, binds) thing_inside
            -- A single non-recursive binding
            -- We want to keep non-recursive things non-recursive
            -- so that we desugar unlifted bindings correctly
      = do { let bind = case bagToList binds of
                     [bind] -> bind
                     []     -> panic "tc_group: empty list of binds"
                     _      -> panic "tc_group: NonRecursive binds is not a singleton bag"
           ; (bind', thing) <- tc_single top_lvl sig_fn prag_fn bind thing_inside
           ; return ( [(NonRecursive, bind')], thing) }

    tc_group top_lvl sig_fn prag_fn (Recursive, binds) thing_inside
      =     -- To maximise polymorphism, we do a new
            -- strongly-connected-component analysis, this time omitting
            -- any references to variables with type signatures.
            -- (This used to be optional, but isn't now.)
        do  { traceTc "tc_group rec" (pprLHsBinds binds)
            ; when hasPatSyn $ recursivePatSynErr binds
            ; (binds1, thing) <- go sccs
            ; return ([(Recursive, binds1)], thing) }
                    -- Rec them all together
      where
        hasPatSyn = anyBag (isPatSyn . unLoc) binds
        isPatSyn PatSynBind{} = True
        isPatSyn _ = False

        sccs :: [SCC (LHsBind Name)]
        sccs = stronglyConnCompFromEdgedVertices (mkEdges sig_fn binds)

        go :: [SCC (LHsBind Name)] -> TcM (LHsBinds TcId, thing)
        go (scc:sccs) = do  { (binds1, ids1, closed) <- tc_scc scc
                            ; (binds2, thing) <- tcExtendLetEnv top_lvl closed ids1 $
                                                 go sccs
                            ; return (binds1 `unionBags` binds2, thing) }
        go []         = do  { thing <- thing_inside; return (emptyBag, thing) }

        tc_scc (AcyclicSCC bind) = tc_sub_group NonRecursive [bind]
        tc_scc (CyclicSCC binds) = tc_sub_group Recursive    binds

        tc_sub_group = tcPolyBinds top_lvl sig_fn prag_fn Recursive

    recursivePatSynErr :: OutputableBndr name => LHsBinds name -> TcM a
    recursivePatSynErr binds
      = failWithTc $
        hang (ptext (sLit "Recursive pattern synonym definition with following bindings:"))
           2 (vcat $ map pprLBind . bagToList $ binds)
      where
        pprLoc loc  = parens (ptext (sLit "defined at") <+> ppr loc)
        pprLBind (L loc bind) = pprWithCommas ppr (collectHsBindBinders bind) <+>
                                pprLoc loc

    tc_single :: forall thing.
                TopLevelFlag -> TcSigFun -> PragFun
              -> LHsBind Name -> TcM thing
              -> TcM (LHsBinds TcId, thing)
    tc_single _top_lvl sig_fn _prag_fn (L _ (PatSynBind psb@PSB{ psb_id = L _ name })) thing_inside
      = do { (pat_syn, aux_binds) <- tc_pat_syn_decl
           ; let tything = AConLike (PatSynCon pat_syn)
           ; thing <- tcExtendGlobalEnv [tything] thing_inside
           ; return (aux_binds, thing)
           }
      where
        tc_pat_syn_decl = case sig_fn name of
            Nothing                  -> tcInferPatSynDecl psb
            Just (TcPatSynInfo tpsi) -> tcCheckPatSynDecl psb tpsi
            Just                  _  -> panic "tc_single"

    tc_single top_lvl sig_fn prag_fn lbind thing_inside
      = do { (binds1, ids, closed) <- tcPolyBinds top_lvl sig_fn prag_fn
                                        NonRecursive NonRecursive
                                        [lbind]
           ; thing <- tcExtendLetEnv top_lvl closed ids thing_inside
           ; return (binds1, thing) }

    -- | No signature or a partial signature
    noCompleteSig :: Maybe TcSigInfo -> Bool
    noCompleteSig Nothing    = True
    noCompleteSig (Just sig) = isPartialSig sig

    ------------------------
    mkEdges :: TcSigFun -> LHsBinds Name -> [Node BKey (LHsBind Name)]

    type BKey = Int -- Just number off the bindings

    mkEdges sig_fn binds
      = [ (bind, key, [key | n <- nameSetElems (bind_fvs (unLoc bind)),
                             Just key <- [lookupNameEnv key_map n], no_sig n ])
        | (bind, key) <- keyd_binds
        ]
      where
        no_sig :: Name -> Bool
        no_sig n = noCompleteSig (sig_fn n)

        keyd_binds = bagToList binds `zip` [0::BKey ..]

        key_map :: NameEnv BKey     -- Which binding it comes from
        key_map = mkNameEnv [(bndr, key) | (L _ bind, key) <- keyd_binds
                                         , bndr <- collectHsBindBinders bind ]

    ------------------------
    tcPolyBinds :: TopLevelFlag -> TcSigFun -> PragFun
                -> RecFlag         -- Whether the group is really recursive
                -> RecFlag         -- Whether it's recursive after breaking
                                   -- dependencies based on type signatures
                -> [LHsBind Name]  -- None are PatSynBind
                -> TcM (LHsBinds TcId, [TcId], TopLevelFlag)

    -- Typechecks a single bunch of values bindings all together,
    -- and generalises them.  The bunch may be only part of a recursive
    -- group, because we use type signatures to maximise polymorphism
    --
    -- Returns a list because the input may be a single non-recursive binding,
    -- in which case the dependency order of the resulting bindings is
    -- important.
    --
    -- Knows nothing about the scope of the bindings
    -- None of the bindings are pattern synonyms

    tcPolyBinds top_lvl sig_fn prag_fn rec_group rec_tc bind_list
      = setSrcSpan loc                              $
        recoverM (recoveryCode binder_names sig_fn) $ do
            -- Set up main recover; take advantage of any type sigs

        { traceTc "------------------------------------------------" Outputable.empty
        ; traceTc "Bindings for {" (ppr binder_names)
        ; dflags   <- getDynFlags
        ; type_env <- getLclTypeEnv
        ; let plan = decideGeneralisationPlan dflags type_env
                             binder_names bind_list sig_fn
        ; traceTc "Generalisation plan" (ppr plan)
        ; result@(tc_binds, poly_ids, _) <- case plan of
             NoGen               -> tcPolyNoGen rec_tc prag_fn sig_fn bind_list
             InferGen mn cl      -> tcPolyInfer rec_tc prag_fn sig_fn mn cl bind_list
             CheckGen lbind sig  -> tcPolyCheck rec_tc prag_fn sig lbind

            -- Check whether strict bindings are ok
            -- These must be non-recursive etc, and are not generalised
            -- They desugar to a case expression in the end
        ; checkStrictBinds top_lvl rec_group bind_list tc_binds poly_ids
        ; traceTc "} End of bindings for" (vcat [ ppr binder_names, ppr rec_group
                                                , vcat [ppr id <+> ppr (idType id) | id <- poly_ids]
                                              ])

        ; return result }
      where
        binder_names = collectHsBindListBinders bind_list
        loc = foldr1 combineSrcSpans (map getLoc bind_list)
             -- The mbinds have been dependency analysed and
             -- may no longer be adjacent; so find the narrowest
             -- span that includes them all

    ------------------
    tcPolyNoGen     -- No generalisation whatsoever
      :: RecFlag       -- Whether it's recursive after breaking
                       -- dependencies based on type signatures
      -> PragFun -> TcSigFun
      -> [LHsBind Name]
      -> TcM (LHsBinds TcId, [TcId], TopLevelFlag)

    tcPolyNoGen rec_tc prag_fn tc_sig_fn bind_list
      = do { (binds', mono_infos) <- tcMonoBinds rec_tc tc_sig_fn
                                                 (LetGblBndr prag_fn)
                                                 bind_list
           ; mono_ids' <- mapM tc_mono_info mono_infos
           ; return (binds', mono_ids', NotTopLevel) }
      where
        tc_mono_info (name, _, mono_id)
          = do { mono_ty' <- zonkTcType (idType mono_id)
                 -- Zonk, mainly to expose unboxed types to checkStrictBinds
               ; let mono_id' = setIdType mono_id mono_ty'
               ; _specs <- tcSpecPrags mono_id' (prag_fn name)
               ; return mono_id' }
               -- NB: tcPrags generates error messages for
               --     specialisation pragmas for non-overloaded sigs
               -- Indeed that is why we call it here!
               -- So we can safely ignore _specs

    ------------------
    tcPolyCheck :: RecFlag       -- Whether it's recursive after breaking
                                 -- dependencies based on type signatures
                -> PragFun -> TcSigInfo
                -> LHsBind Name
                -> TcM (LHsBinds TcId, [TcId], TopLevelFlag)
    -- There is just one binding,
    --   it binds a single variable,
    --   it has a signature,
    tcPolyCheck rec_tc prag_fn
                sig@(TcSigInfo { sig_id = poly_id, sig_tvs = tvs_w_scoped
                               , sig_nwcs = sig_nwcs, sig_theta = theta
                               , sig_tau = tau, sig_loc = loc })
                bind
      = ASSERT( null sig_nwcs ) -- We should be in tcPolyInfer if there are wildcards
        do { ev_vars <- newEvVars theta
           ; let skol_info = SigSkol (FunSigCtxt (idName poly_id)) (mkPhiTy theta tau)
                 prag_sigs = prag_fn (idName poly_id)
                 tvs = map snd tvs_w_scoped
           ; (ev_binds, (binds', [mono_info]))
                <- setSrcSpan loc $
                   checkConstraints skol_info tvs ev_vars $
                   tcMonoBinds rec_tc (\_ -> Just sig) LetLclBndr [bind]

           ; spec_prags <- tcSpecPrags poly_id prag_sigs
           ; poly_id    <- addInlinePrags poly_id prag_sigs

           ; let (_, _, mono_id) = mono_info
                 export = ABE { abe_wrap = idHsWrapper
                              , abe_poly = poly_id
                              , abe_mono = mono_id
                              , abe_prags = SpecPrags spec_prags }
                 abs_bind = L loc $ AbsBinds
                            { abs_tvs = tvs
                            , abs_ev_vars = ev_vars, abs_ev_binds = ev_binds
                            , abs_exports = [export], abs_binds = binds' }
                 closed | isEmptyVarSet (tyVarsOfType (idType poly_id)) = TopLevel
                        | otherwise                                     = NotTopLevel
           ; return (unitBag abs_bind, [poly_id], closed) }

    tcPolyCheck _rec_tc _prag_fn sig _bind
      = pprPanic "tcPolyCheck" (ppr sig)

    ------------------
    tcPolyInfer
      :: RecFlag       -- Whether it's recursive after breaking
                       -- dependencies based on type signatures
      -> PragFun -> TcSigFun
      -> Bool         -- True <=> apply the monomorphism restriction
      -> Bool         -- True <=> free vars have closed types
      -> [LHsBind Name]
      -> TcM (LHsBinds TcId, [TcId], TopLevelFlag)
    tcPolyInfer rec_tc prag_fn tc_sig_fn mono closed bind_list
      = do { (((binds', mono_infos), tclvl), wanted)
                 <- captureConstraints  $
                    captureTcLevel      $
                    tcMonoBinds rec_tc tc_sig_fn LetLclBndr bind_list

           ; let name_taus = [(name, idType mono_id) | (name, _, mono_id) <- mono_infos]
           ; traceTc "simplifyInfer call" (ppr name_taus $$ ppr wanted)
           ; (qtvs, givens, mr_bites, ev_binds)
                     <- simplifyInfer tclvl mono name_taus wanted

           ; inferred_theta  <- zonkTcThetaType (map evVarPred givens)
           ; exports <- checkNoErrs $ mapM (mkExport prag_fn qtvs inferred_theta)
                                           mono_infos

           ; loc <- getSrcSpanM
           ; let poly_ids = map abe_poly exports
                 final_closed | closed && not mr_bites = TopLevel
                              | otherwise              = NotTopLevel
                 abs_bind = L loc $
                            AbsBinds { abs_tvs = qtvs
                                     , abs_ev_vars = givens, abs_ev_binds = ev_binds
                                     , abs_exports = exports, abs_binds = binds' }

           ; traceTc "Binding:" (ppr final_closed $$
                                 ppr (poly_ids `zip` map idType poly_ids))
           ; return (unitBag abs_bind, poly_ids, final_closed) }
             -- poly_ids are guaranteed zonked by mkExport

    --------------
    mkExport :: PragFun
             -> [TyVar] -> TcThetaType      -- Both already zonked
             -> MonoBindInfo
             -> TcM (ABExport Id)
    -- Only called for generalisation plan IferGen, not by CheckGen or NoGen
    --
    -- mkExport generates exports with
    --      zonked type variables,
    --      zonked poly_ids
    -- The former is just because no further unifications will change
    -- the quantified type variables, so we can fix their final form
    -- right now.
    -- The latter is needed because the poly_ids are used to extend the
    -- type environment; see the invariant on TcEnv.tcExtendIdEnv

    -- Pre-condition: the qtvs and theta are already zonked

    mkExport prag_fn qtvs inferred_theta (poly_name, mb_sig, mono_id)
      = do  { mono_ty <- zonkTcType (idType mono_id)

            ; poly_id <- case mb_sig of
                  Nothing -> mkInferredPolyId poly_name qtvs inferred_theta mono_ty
                  Just (TcPatSynInfo _) -> panic "mkExport"
                  Just sig | isPartialSig sig
                           -> do { final_theta <- completeTheta inferred_theta sig
                                 ; mkInferredPolyId poly_name qtvs final_theta mono_ty }
                           | otherwise
                           -> return (sig_id sig)

            -- NB: poly_id has a zonked type
            ; poly_id <- addInlinePrags poly_id prag_sigs
            ; spec_prags <- tcSpecPrags poly_id prag_sigs
                    -- tcPrags requires a zonked poly_id

            ; let sel_poly_ty = mkSigmaTy qtvs inferred_theta mono_ty
            ; traceTc "mkExport: check sig"
                      (vcat [ ppr poly_name, ppr sel_poly_ty, ppr (idType poly_id) ])

            -- Perform the impedence-matching and ambiguity check
            -- right away.  If it fails, we want to fail now (and recover
            -- in tcPolyBinds).  If we delay checking, we get an error cascade.
            -- Remember we are in the tcPolyInfer case, so the type envt is
            -- closed (unless we are doing NoMonoLocalBinds in which case all bets
            -- are off)
            -- See Note [Impedence matching]
            ; (wrap, wanted) <- addErrCtxtM (mk_bind_msg inferred True poly_name (idType poly_id)) $
                                captureConstraints $
                                tcSubType_NC sig_ctxt sel_poly_ty (idType poly_id)
            ; ev_binds <- simplifyTop wanted

            ; return (ABE { abe_wrap = mkWpLet (EvBinds ev_binds) <.> wrap
                          , abe_poly = poly_id
                          , abe_mono = mono_id
                          , abe_prags = SpecPrags spec_prags }) }
      where
        inferred = isNothing mb_sig
        prag_sigs = prag_fn poly_name
        sig_ctxt  = InfSigCtxt poly_name

    mkInferredPolyId :: Name -> [TyVar] -> TcThetaType -> TcType -> TcM Id
    -- In the inference case (no signature) this stuff figures out
    -- the right type variables and theta to quantify over
    -- See Note [Validity of inferred types]
    mkInferredPolyId poly_name qtvs theta mono_ty
      = do { fam_envs <- tcGetFamInstEnvs

           ; let (_co, norm_mono_ty) = normaliseType fam_envs Nominal mono_ty
                   -- Unification may not have normalised the type,
                   -- (see Note [Lazy flattening] in TcFlatten) so do it
                   -- here to make it as uncomplicated as possible.
                   -- Example: f :: [F Int] -> Bool
                   -- should be rewritten to f :: [Char] -> Bool, if possible

                 my_tvs2 = closeOverKinds (growThetaTyVars theta (tyVarsOfType norm_mono_ty))
                      -- Include kind variables!  Trac #7916

                 my_tvs   = filter (`elemVarSet` my_tvs2) qtvs   -- Maintain original order
                 my_theta = filter (quantifyPred my_tvs2) theta
                 inferred_poly_ty = mkSigmaTy my_tvs my_theta norm_mono_ty

           ; addErrCtxtM (mk_bind_msg True False poly_name inferred_poly_ty) $
             checkValidType (InfSigCtxt poly_name) inferred_poly_ty

           ; return (mkLocalId poly_name inferred_poly_ty) }

    mk_bind_msg :: Bool -> Bool -> Name -> TcType -> TidyEnv -> TcM (TidyEnv, SDoc)
    mk_bind_msg inferred want_ambig poly_name poly_ty tidy_env
     = do { (tidy_env', tidy_ty) <- zonkTidyTcType tidy_env poly_ty
          ; return (tidy_env', mk_msg tidy_ty) }
     where
       mk_msg ty = vcat [ ptext (sLit "When checking that") <+> quotes (ppr poly_name)
                          <+> ptext (sLit "has the") <+> what <+> ptext (sLit "type")
                        , nest 2 (ppr poly_name <+> dcolon <+> ppr ty)
                        , ppWhen want_ambig $
                          ptext (sLit "Probable cause: the inferred type is ambiguous") ]
       what | inferred  = ptext (sLit "inferred")
            | otherwise = ptext (sLit "specified")

    -- | Report the inferred constraints for an extra-constraints wildcard/hole as
    -- an error message, unless the PartialTypeSignatures flag is enabled. In this
    -- case, the extra inferred constraints are accepted without complaining.
    -- Returns the annotated constraints combined with the inferred constraints.
    completeTheta :: TcThetaType -> TcSigInfo -> TcM TcThetaType
    completeTheta _ (TcPatSynInfo _)
      = panic "Extra-constraints wildcard not supported in a pattern signature"
    completeTheta inferred_theta
                  sig@(TcSigInfo { sig_id = poly_id
                                 , sig_extra_cts = mb_extra_cts
                                 , sig_theta = annotated_theta })
      | Just loc <- mb_extra_cts
      = do { annotated_theta <- zonkTcThetaType annotated_theta
           ; let inferred_diff = minusList inferred_theta annotated_theta
                 final_theta   = annotated_theta ++ inferred_diff
           ; partial_sigs      <- xoptM Opt_PartialTypeSignatures
           ; warn_partial_sigs <- woptM Opt_WarnPartialTypeSignatures
           ; msg <- mkLongErrAt loc (mk_msg inferred_diff partial_sigs) empty
           ; case partial_sigs of
               True | warn_partial_sigs -> reportWarning $ makeIntoWarning msg
                    | otherwise         -> return ()
               False                    -> reportError msg
           ; return final_theta }

      | otherwise
      = zonkTcThetaType annotated_theta
        -- No extra-constraints wildcard means no extra constraints will be added
        -- to the context, so just return the possibly empty (zonked)
        -- annotated_theta.
      where
        pts_hint = text "To use the inferred type, enable PartialTypeSignatures"
        mk_msg inferred_diff suppress_hint
           = vcat [ hang ((text "Found hole") <+> quotes (char '_'))
                       2 (text "with inferred constraints:")
                          <+> pprTheta inferred_diff
                  , if suppress_hint then empty else pts_hint
                  , typeSigCtxt (idName poly_id) sig ]

    {-
    Note [Validity of inferred types]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We need to check inferred type for validity, in case it uses language
    extensions that are not turned on.  The principle is that if the user
    simply adds the inferred type to the program source, it'll compile fine.
    See #8883.

    Examples that might fail:
     - an inferred theta that requires type equalities e.g. (F a ~ G b)
                                    or multi-parameter type classes
     - an inferred type that includes unboxed tuples

    However we don't do the ambiguity check (checkValidType omits it for
    InfSigCtxt) because the impedence-matching stage, which follows
    immediately, will do it and we don't want two error messages.
    Moreover, because of the impedence matching stage, the ambiguity-check
    suggestion of -XAllowAmbiguiousTypes will not work.

    Note [Impedence matching]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider
       f 0 x = x
       f n x = g [] (not x)

       g [] y = f 10 y
       g _  y = f 9  y

    After typechecking we'll get
      f_mono_ty :: a -> Bool -> Bool
      g_mono_ty :: [b] -> Bool -> Bool
    with constraints
      (Eq a, Num a)

    Note that f is polymorphic in 'a' and g in 'b'; and these are not linked.
    The types we really want for f and g are
       f :: forall a. (Eq a, Num a) => a -> Bool -> Bool
       g :: forall b. [b] -> Bool -> Bool

    We can get these by "impedence matching":
       tuple :: forall a b. (Eq a, Num a) => (a -> Bool -> Bool, [b] -> Bool -> Bool)
       tuple a b d1 d1 = let ...bind f_mono, g_mono in (f_mono, g_mono)

       f a d1 d2 = case tuple a Any d1 d2 of (f, g) -> f
       g b = case tuple Integer b dEqInteger dNumInteger of (f,g) -> g

    Suppose the shared quantified tyvars are qtvs and constraints theta.
    Then we want to check that
       f's polytype  is more polymorphic than   forall qtvs. theta => f_mono_ty
    and the proof is the impedence matcher.

    Notice that the impedence matcher may do defaulting.  See Trac #7173.

    It also cleverly does an ambiguity check; for example, rejecting
       f :: F a -> a
    where F is a non-injective type function.
    -}

    type PragFun = Name -> [LSig Name]

    mkPragFun :: [LSig Name] -> LHsBinds Name -> PragFun
    mkPragFun sigs binds = \n -> lookupNameEnv prag_env n `orElse` []
      where
        prs = mapMaybe get_sig sigs

        get_sig :: LSig Name -> Maybe (Located Name, LSig Name)
        get_sig (L l (SpecSig nm ty inl)) = Just (nm, L l $ SpecSig  nm ty (add_arity nm inl))
        get_sig (L l (InlineSig nm inl))  = Just (nm, L l $ InlineSig nm   (add_arity nm inl))
        get_sig _                         = Nothing

        add_arity (L _ n) inl_prag   -- Adjust inl_sat field to match visible arity of function
          | Just ar <- lookupNameEnv ar_env n,
            Inline <- inl_inline inl_prag     = inl_prag { inl_sat = Just ar }
            -- add arity only for real INLINE pragmas, not INLINABLE
          | otherwise                         = inl_prag

        prag_env :: NameEnv [LSig Name]
        prag_env = foldl add emptyNameEnv prs
        add env (L _ n,p) = extendNameEnv_Acc (:) singleton env n p

        -- ar_env maps a local to the arity of its definition
        ar_env :: NameEnv Arity
        ar_env = foldrBag lhsBindArity emptyNameEnv binds

    lhsBindArity :: LHsBind Name -> NameEnv Arity -> NameEnv Arity
    lhsBindArity (L _ (FunBind { fun_id = id, fun_matches = ms })) env
      = extendNameEnv env (unLoc id) (matchGroupArity ms)
    lhsBindArity _ env = env        -- PatBind/VarBind

    ------------------
    tcSpecPrags :: Id -> [LSig Name]
                -> TcM [LTcSpecPrag]
    -- Add INLINE and SPECIALSE pragmas
    --    INLINE prags are added to the (polymorphic) Id directly
    --    SPECIALISE prags are passed to the desugarer via TcSpecPrags
    -- Pre-condition: the poly_id is zonked
    -- Reason: required by tcSubExp
    tcSpecPrags poly_id prag_sigs
      = do { traceTc "tcSpecPrags" (ppr poly_id <+> ppr spec_sigs)
           ; unless (null bad_sigs) warn_discarded_sigs
           ; pss <- mapAndRecoverM (wrapLocM (tcSpec poly_id)) spec_sigs
           ; return $ concatMap (\(L l ps) -> map (L l) ps) pss }
      where
        spec_sigs = filter isSpecLSig prag_sigs
        bad_sigs  = filter is_bad_sig prag_sigs
        is_bad_sig s = not (isSpecLSig s || isInlineLSig s)

        warn_discarded_sigs = warnPrags poly_id bad_sigs $
                              ptext (sLit "Discarding unexpected pragmas for")

    --------------
    tcSpec :: TcId -> Sig Name -> TcM [TcSpecPrag]
    tcSpec poly_id prag@(SpecSig fun_name hs_tys inl)
      -- The Name fun_name in the SpecSig may not be the same as that of the poly_id
      -- Example: SPECIALISE for a class method: the Name in the SpecSig is
      --          for the selector Id, but the poly_id is something like $cop
      -- However we want to use fun_name in the error message, since that is
      -- what the user wrote (Trac #8537)
      = addErrCtxt (spec_ctxt prag) $
        do  { spec_tys <- mapM (tcHsSigType sig_ctxt) hs_tys
            ; warnIf (not (isOverloadedTy poly_ty || isInlinePragma inl))
                     (ptext (sLit "SPECIALISE pragma for non-overloaded function")
                      <+> quotes (ppr fun_name))
                      -- Note [SPECIALISE pragmas]
            ; wraps <- mapM (tcSubType sig_ctxt (idType poly_id)) spec_tys
            ; return [ (SpecPrag poly_id wrap inl) | wrap <- wraps ] }
      where
        name      = idName poly_id
        poly_ty   = idType poly_id
        sig_ctxt  = FunSigCtxt name
        spec_ctxt prag = hang (ptext (sLit "In the SPECIALISE pragma")) 2 (ppr prag)

    tcSpec _ prag = pprPanic "tcSpec" (ppr prag)

    --------------
    tcImpPrags :: [LSig Name] -> TcM [LTcSpecPrag]
    -- SPECIALISE pragmas for imported things
    tcImpPrags prags
      = do { this_mod <- getModule
           ; dflags <- getDynFlags
           ; if (not_specialising dflags) then
                return []
             else do
                { pss <- mapAndRecoverM (wrapLocM tcImpSpec)
                         [L loc (name,prag)
                                   | (L loc prag@(SpecSig (L _ name) _ _)) <- prags
                                   , not (nameIsLocalOrFrom this_mod name) ]
                ; return $ concatMap (\(L l ps) -> map (L l) ps) pss } }
      where
        -- Ignore SPECIALISE pragmas for imported things
        -- when we aren't specialising, or when we aren't generating
        -- code.  The latter happens when Haddocking the base library;
        -- we don't wnat complaints about lack of INLINABLE pragmas
        not_specialising dflags
          | not (gopt Opt_Specialise dflags) = True
          | otherwise = case hscTarget dflags of
                          HscNothing -> True
                          HscInterpreted -> True
                          _other         -> False

    tcImpSpec :: (Name, Sig Name) -> TcM [TcSpecPrag]
    tcImpSpec (name, prag)
     = do { id <- tcLookupId name
          ; unless (isAnyInlinePragma (idInlinePragma id))
                   (addWarnTc (impSpecErr name))
          ; tcSpec id prag }

    impSpecErr :: Name -> SDoc
    impSpecErr name
      = hang (ptext (sLit "You cannot SPECIALISE") <+> quotes (ppr name))
           2 (vcat [ ptext (sLit "because its definition has no INLINE/INLINABLE pragma")
                   , parens $ sep
                       [ ptext (sLit "or its defining module") <+> quotes (ppr mod)
                       , ptext (sLit "was compiled without -O")]])
      where
        mod = nameModule name

    --------------
    tcVectDecls :: [LVectDecl Name] -> TcM ([LVectDecl TcId])
    tcVectDecls decls
      = do { decls' <- mapM (wrapLocM tcVect) decls
           ; let ids  = [lvectDeclName decl | decl <- decls', not $ lvectInstDecl decl]
                 dups = findDupsEq (==) ids
           ; mapM_ reportVectDups dups
           ; traceTcConstraints "End of tcVectDecls"
           ; return decls'
           }
      where
        reportVectDups (first:_second:_more)
          = addErrAt (getSrcSpan first) $
              ptext (sLit "Duplicate vectorisation declarations for") <+> ppr first
        reportVectDups _ = return ()

    --------------
    tcVect :: VectDecl Name -> TcM (VectDecl TcId)
    -- FIXME: We can't typecheck the expression of a vectorisation declaration against the vectorised
    --   type of the original definition as this requires internals of the vectoriser not available
    --   during type checking.  Instead, constrain the rhs of a vectorisation declaration to be a single
    --   identifier (this is checked in 'rnHsVectDecl').  Fix this by enabling the use of 'vectType'
    --   from the vectoriser here.
    tcVect (HsVect s name rhs)
      = addErrCtxt (vectCtxt name) $
        do { var <- wrapLocM tcLookupId name
           ; let L rhs_loc (HsVar rhs_var_name) = rhs
           ; rhs_id <- tcLookupId rhs_var_name
           ; return $ HsVect s var (L rhs_loc (HsVar rhs_id))
           }

    {- OLD CODE:
             -- turn the vectorisation declaration into a single non-recursive binding
           ; let bind    = L loc $ mkTopFunBind name [mkSimpleMatch [] rhs]
                 sigFun  = const Nothing
                 pragFun = mkPragFun [] (unitBag bind)

             -- perform type inference (including generalisation)
           ; (binds, [id'], _) <- tcPolyInfer False True sigFun pragFun NonRecursive [bind]

           ; traceTc "tcVect inferred type" $ ppr (varType id')
           ; traceTc "tcVect bindings"      $ ppr binds

             -- add all bindings, including the type variable and dictionary bindings produced by type
             -- generalisation to the right-hand side of the vectorisation declaration
           ; let [AbsBinds tvs evs _ evBinds actualBinds] = (map unLoc . bagToList) binds
           ; let [bind']                                  = bagToList actualBinds
                 MatchGroup
                   [L _ (Match _ _ (GRHSs [L _ (GRHS _ rhs')] _))]
                   _                                      = (fun_matches . unLoc) bind'
                 rhsWrapped                               = mkHsLams tvs evs (mkHsDictLet evBinds rhs')

            -- We return the type-checked 'Id', to propagate the inferred signature
            -- to the vectoriser - see "Note [Typechecked vectorisation pragmas]" in HsDecls
           ; return $ HsVect (L loc id') (Just rhsWrapped)
           }
     -}
    tcVect (HsNoVect s name)
      = addErrCtxt (vectCtxt name) $
        do { var <- wrapLocM tcLookupId name
           ; return $ HsNoVect s var
           }
    tcVect (HsVectTypeIn _ isScalar lname rhs_name)
      = addErrCtxt (vectCtxt lname) $
        do { tycon <- tcLookupLocatedTyCon lname
           ; checkTc (   not isScalar             -- either    we have a non-SCALAR declaration
                     || isJust rhs_name           -- or        we explicitly provide a vectorised type
                     || tyConArity tycon == 0     -- otherwise the type constructor must be nullary
                     )
                     scalarTyConMustBeNullary

           ; rhs_tycon <- fmapMaybeM (tcLookupTyCon . unLoc) rhs_name
           ; return $ HsVectTypeOut isScalar tycon rhs_tycon
           }
    tcVect (HsVectTypeOut _ _ _)
      = panic "TcBinds.tcVect: Unexpected 'HsVectTypeOut'"
    tcVect (HsVectClassIn _ lname)
      = addErrCtxt (vectCtxt lname) $
        do { cls <- tcLookupLocatedClass lname
           ; return $ HsVectClassOut cls
           }
    tcVect (HsVectClassOut _)
      = panic "TcBinds.tcVect: Unexpected 'HsVectClassOut'"
    tcVect (HsVectInstIn linstTy)
      = addErrCtxt (vectCtxt linstTy) $
        do { (cls, tys) <- tcHsVectInst linstTy
           ; inst       <- tcLookupInstance cls tys
           ; return $ HsVectInstOut inst
           }
    tcVect (HsVectInstOut _)
      = panic "TcBinds.tcVect: Unexpected 'HsVectInstOut'"

    vectCtxt :: Outputable thing => thing -> SDoc
    vectCtxt thing = ptext (sLit "When checking the vectorisation declaration for") <+> ppr thing

    scalarTyConMustBeNullary :: MsgDoc
    scalarTyConMustBeNullary = ptext (sLit "VECTORISE SCALAR type constructor must be nullary")

    --------------
    -- If typechecking the binds fails, then return with each
    -- signature-less binder given type (forall a.a), to minimise
    -- subsequent error messages
    recoveryCode :: [Name] -> TcSigFun -> TcM (LHsBinds TcId, [Id], TopLevelFlag)
    recoveryCode binder_names sig_fn
      = do  { traceTc "tcBindsWithSigs: error recovery" (ppr binder_names)
            ; poly_ids <- mapM mk_dummy binder_names
            ; return (emptyBag, poly_ids, if all is_closed poly_ids
                                          then TopLevel else NotTopLevel) }
      where
        mk_dummy name
            | isJust (sig_fn name) = tcLookupId name        -- Had signature; look it up
            | otherwise            = return (mkLocalId name forall_a_a)    -- No signature

        is_closed poly_id = isEmptyVarSet (tyVarsOfType (idType poly_id))

    forall_a_a :: TcType
    forall_a_a = mkForAllTy openAlphaTyVar (mkTyVarTy openAlphaTyVar)

    {-
    Note [SPECIALISE pragmas]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    There is no point in a SPECIALISE pragma for a non-overloaded function:
       reverse :: [a] -> [a]
       {-# SPECIALISE reverse :: [Int] -> [Int] #-}

    But SPECIALISE INLINE *can* make sense for GADTS:
       data Arr e where
         ArrInt :: !Int -> ByteArray# -> Arr Int
         ArrPair :: !Int -> Arr e1 -> Arr e2 -> Arr (e1, e2)

       (!:) :: Arr e -> Int -> e
       {-# SPECIALISE INLINE (!:) :: Arr Int -> Int -> Int #-}
       {-# SPECIALISE INLINE (!:) :: Arr (a, b) -> Int -> (a, b) #-}
       (ArrInt _ ba)     !: (I# i) = I# (indexIntArray# ba i)
       (ArrPair _ a1 a2) !: i      = (a1 !: i, a2 !: i)

    When (!:) is specialised it becomes non-recursive, and can usefully
    be inlined.  Scary!  So we only warn for SPECIALISE *without* INLINE
    for a non-overloaded function.

    ************************************************************************
    *                                                                      *
    \subsection{tcMonoBind}
    *                                                                      *
    ************************************************************************

    @tcMonoBinds@ deals with a perhaps-recursive group of HsBinds.
    The signatures have been dealt with already.

    Note [Pattern bindings]
    ~~~~~~~~~~~~~~~~~~~~~~~
    The rule for typing pattern bindings is this:

        ..sigs..
        p = e

    where 'p' binds v1..vn, and 'e' may mention v1..vn,
    typechecks exactly like

        ..sigs..
        x = e       -- Inferred type
        v1 = case x of p -> v1
        ..
        vn = case x of p -> vn

    Note that
        (f :: forall a. a -> a) = id
    should not typecheck because
           case id of { (f :: forall a. a->a) -> f }
    will not typecheck.
    -}

    tcMonoBinds :: RecFlag  -- Whether the binding is recursive for typechecking purposes
                            -- i.e. the binders are mentioned in their RHSs, and
                            --      we are not rescued by a type signature
                -> TcSigFun -> LetBndrSpec
                -> [LHsBind Name]
                -> TcM (LHsBinds TcId, [MonoBindInfo])

    tcMonoBinds is_rec sig_fn no_gen
               [ L b_loc (FunBind { fun_id = L nm_loc name, fun_infix = inf,
                                    fun_matches = matches, bind_fvs = fvs })]
                                 -- Single function binding,
      | NonRecursive <- is_rec   -- ...binder isn't mentioned in RHS
      , Nothing <- sig_fn name   -- ...with no type signature
      =     -- In this very special case we infer the type of the
            -- right hand side first (it may have a higher-rank type)
            -- and *then* make the monomorphic Id for the LHS
            -- e.g.         f = \(x::forall a. a->a) -> <body>
            --      We want to infer a higher-rank type for f
        setSrcSpan b_loc    $
        do  { rhs_ty  <- newFlexiTyVarTy openTypeKind
            ; mono_id <- newNoSigLetBndr no_gen name rhs_ty
            ; (co_fn, matches') <- tcExtendIdBndrs [TcIdBndr mono_id NotTopLevel] $
                                     -- We extend the error context even for a non-recursive
                                     -- function so that in type error messages we show the
                                     -- type of the thing whose rhs we are type checking
                                   tcMatchesFun name inf matches rhs_ty

            ; return (unitBag $ L b_loc $
                         FunBind { fun_id = L nm_loc mono_id, fun_infix = inf,
                                   fun_matches = matches', bind_fvs = fvs,
                                   fun_co_fn = co_fn, fun_tick = [] },
                      [(name, Nothing, mono_id)]) }

    tcMonoBinds _ sig_fn no_gen binds
      = do  { tc_binds <- mapM (wrapLocM (tcLhs sig_fn no_gen)) binds

            -- Bring the monomorphic Ids, into scope for the RHSs
            ; let mono_info  = getMonoBindInfo tc_binds
                  rhs_id_env = [(name, mono_id) | (name, mb_sig, mono_id) <- mono_info
                                                , noCompleteSig mb_sig ]
                        -- A monomorphic binding for each term variable that lacks
                        -- a type sig.  (Ones with a sig are already in scope.)

            ; traceTc "tcMonoBinds" $ vcat [ ppr n <+> ppr id <+> ppr (idType id)
                                           | (n,id) <- rhs_id_env]
            ; binds' <- tcExtendIdEnv2 rhs_id_env $
                        mapM (wrapLocM tcRhs) tc_binds
            ; return (listToBag binds', mono_info) }

    ------------------------
    -- tcLhs typechecks the LHS of the bindings, to construct the environment in which
    -- we typecheck the RHSs.  Basically what we are doing is this: for each binder:
    --      if there's a signature for it, use the instantiated signature type
    --      otherwise invent a type variable
    -- You see that quite directly in the FunBind case.
    --
    -- But there's a complication for pattern bindings:
    --      data T = MkT (forall a. a->a)
    --      MkT f = e
    -- Here we can guess a type variable for the entire LHS (which will be refined to T)
    -- but we want to get (f::forall a. a->a) as the RHS environment.
    -- The simplest way to do this is to typecheck the pattern, and then look up the
    -- bound mono-ids.  Then we want to retain the typechecked pattern to avoid re-doing
    -- it; hence the TcMonoBind data type in which the LHS is done but the RHS isn't

    data TcMonoBind         -- Half completed; LHS done, RHS not done
      = TcFunBind  MonoBindInfo  SrcSpan Bool (MatchGroup Name (LHsExpr Name))
      | TcPatBind [MonoBindInfo] (LPat TcId) (GRHSs Name (LHsExpr Name)) TcSigmaType

    type MonoBindInfo = (Name, Maybe TcSigInfo, TcId)
            -- Type signature (if any), and
            -- the monomorphic bound things

    tcLhs :: TcSigFun -> LetBndrSpec -> HsBind Name -> TcM TcMonoBind
    tcLhs sig_fn no_gen (FunBind { fun_id = L nm_loc name, fun_infix = inf, fun_matches = matches })
      | Just sig <- sig_fn name
      = ASSERT2( case no_gen of { LetLclBndr -> True; LetGblBndr {} -> False }
               , ppr name )  -- { f :: ty; f x = e } is always done via CheckGen
                             -- which gives rise to LetLclBndr.  It wouldn't make
                             -- sense to have a *polymorphic* function Id at this point
        do  { mono_name <- newLocalName name
            ; let mono_id = mkLocalId mono_name (sig_tau sig)
            ; addErrCtxt (typeSigCtxt name sig) $
              emitWildcardHoleConstraints (sig_nwcs sig)
            ; return (TcFunBind (name, Just sig, mono_id) nm_loc inf matches) }
      | otherwise
      = do  { mono_ty <- newFlexiTyVarTy openTypeKind
            ; mono_id <- newNoSigLetBndr no_gen name mono_ty
            ; return (TcFunBind (name, Nothing, mono_id) nm_loc inf matches) }

    -- TODOT: emit Hole Constraints for wildcards
    tcLhs sig_fn no_gen (PatBind { pat_lhs = pat, pat_rhs = grhss })
      = do  { let tc_pat exp_ty = tcLetPat sig_fn no_gen pat exp_ty $
                                  mapM lookup_info (collectPatBinders pat)

                    -- After typechecking the pattern, look up the binder
                    -- names, which the pattern has brought into scope.
                  lookup_info :: Name -> TcM MonoBindInfo
                  lookup_info name = do { mono_id <- tcLookupId name
                                        ; return (name, sig_fn name, mono_id) }

            ; ((pat', infos), pat_ty) <- addErrCtxt (patMonoBindsCtxt pat grhss) $
                                         tcInfer tc_pat

            ; return (TcPatBind infos pat' grhss pat_ty) }

    tcLhs _ _ other_bind = pprPanic "tcLhs" (ppr other_bind)
            -- AbsBind, VarBind impossible

    -------------------
    tcRhs :: TcMonoBind -> TcM (HsBind TcId)
    -- When we are doing pattern bindings, or multiple function bindings at a time
    -- we *don't* bring any scoped type variables into scope
    -- Wny not?  They are not completely rigid.
    -- That's why we have the special case for a single FunBind in tcMonoBinds
    tcRhs (TcFunBind (_, mb_sig, mono_id) loc inf matches)
      = tcExtendIdBndrs [TcIdBndr mono_id NotTopLevel] $
        tcExtendTyVarEnv2 tvsAndNwcs $
                -- NotTopLevel: it's a monomorphic binding
        do  { traceTc "tcRhs: fun bind" (ppr mono_id $$ ppr (idType mono_id))
            ; (co_fn, matches') <- tcMatchesFun (idName mono_id) inf
                                                matches (idType mono_id)
            ; return (FunBind { fun_id = L loc mono_id, fun_infix = inf
                              , fun_matches = matches'
                              , fun_co_fn = co_fn
                              , bind_fvs = placeHolderNamesTc
                              , fun_tick = [] }) }
        where
          tvsAndNwcs = maybe [] (\sig -> [(n, tv) | (Just n, tv) <- sig_tvs sig]
                                         ++ sig_nwcs sig) mb_sig

    tcRhs (TcPatBind infos pat' grhss pat_ty)
      = tcExtendIdBndrs [ TcIdBndr mono_id NotTopLevel | (_,_,mono_id) <- infos ] $
                -- NotTopLevel: it's a monomorphic binding
        do  { traceTc "tcRhs: pat bind" (ppr pat' $$ ppr pat_ty)
            ; grhss' <- addErrCtxt (patMonoBindsCtxt pat' grhss) $
                        tcGRHSsPat grhss pat_ty
            ; return (PatBind { pat_lhs = pat', pat_rhs = grhss', pat_rhs_ty = pat_ty
                              , bind_fvs = placeHolderNamesTc
                              , pat_ticks = ([],[]) }) }

    ---------------------
    getMonoBindInfo :: [Located TcMonoBind] -> [MonoBindInfo]
    getMonoBindInfo tc_binds
      = foldr (get_info . unLoc) [] tc_binds
      where
        get_info (TcFunBind info _ _ _)  rest = info : rest
        get_info (TcPatBind infos _ _ _) rest = infos ++ rest

    {-
    ************************************************************************
    *                                                                      *
                    Signatures
    *                                                                      *
    ************************************************************************

    Type signatures are tricky.  See Note [Signature skolems] in TcType

    @tcSigs@ checks the signatures for validity, and returns a list of
    {\em freshly-instantiated} signatures.  That is, the types are already
    split up, and have fresh type variables installed.  All non-type-signature
    "RenamedSigs" are ignored.

    The @TcSigInfo@ contains @TcTypes@ because they are unified with
    the variable's type, and after that checked to see whether they've
    been instantiated.

    Note [Scoped tyvars]
    ~~~~~~~~~~~~~~~~~~~~
    The -XScopedTypeVariables flag brings lexically-scoped type variables
    into scope for any explicitly forall-quantified type variables:
            f :: forall a. a -> a
            f x = e
    Then 'a' is in scope inside 'e'.

    However, we do *not* support this
      - For pattern bindings e.g
            f :: forall a. a->a
            (f,g) = e

    Note [Signature skolems]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    When instantiating a type signature, we do so with either skolems or
    SigTv meta-type variables depending on the use_skols boolean.  This
    variable is set True when we are typechecking a single function
    binding; and False for pattern bindings and a group of several
    function bindings.

    Reason: in the latter cases, the "skolems" can be unified together,
            so they aren't properly rigid in the type-refinement sense.
    NB: unless we are doing H98, each function with a sig will be done
        separately, even if it's mutually recursive, so use_skols will be True

    Note [Only scoped tyvars are in the TyVarEnv]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We are careful to keep only the *lexically scoped* type variables in
    the type environment.  Why?  After all, the renamer has ensured
    that only legal occurrences occur, so we could put all type variables
    into the type env.

    But we want to check that two distinct lexically scoped type variables
    do not map to the same internal type variable.  So we need to know which
    the lexically-scoped ones are... and at the moment we do that by putting
    only the lexically scoped ones into the environment.

    Note [Instantiate sig with fresh variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    It's vital to instantiate a type signature with fresh variables.
    For example:
          type T = forall a. [a] -> [a]
          f :: T;
          f = g where { g :: T; g = <rhs> }

     We must not use the same 'a' from the defn of T at both places!!
    (Instantiation is only necessary because of type synonyms.  Otherwise,
    it's all cool; each signature has distinct type variables from the renamer.)

    Note [Fail eagerly on bad signatures]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    If a type signaure is wrong, fail immediately:

     * the type sigs may bind type variables, so proceeding without them
       can lead to a cascade of errors

     * the type signature might be ambiguous, in which case checking
       the code against the signature will give a very similar error
       to the ambiguity error.

    ToDo: this means we fall over if any type sig
    is wrong (eg at the top level of the module),
    which is over-conservative
    -}

    tcTySigs :: [LSig Name] -> TcM ([TcId], TcSigFun, [TcTyVar])
    tcTySigs hs_sigs
      = checkNoErrs $   -- See Note [Fail eagerly on bad signatures]
        do { (ty_sigs_s, tyvarsl) <- unzip <$> mapAndRecoverM tcTySig hs_sigs
           ; let ty_sigs = concat ty_sigs_s
                 poly_ids = [id | TcSigInfo{ sig_id = id } <- ty_sigs]
                 env = mkNameEnv [(getName sig, sig) | sig <- ty_sigs]
           ; return (poly_ids, lookupNameEnv env, concat tyvarsl) }

    tcTySig :: LSig Name -> TcM ([TcSigInfo], [TcTyVar])
    tcTySig (L _ (IdSig id))
      = do { sig <- instTcTySigFromId id
           ; return ([sig], []) }
    tcTySig (L loc (TypeSig names@(L _ name1 : _) hs_ty wcs))
      = setSrcSpan loc $
        pushTcLevelM   $
        do { nwc_tvs <- mapM newWildcardVarMetaKind wcs      -- Generate fresh meta vars for the wildcards
           ; sigma_ty <- tcExtendTyVarEnv nwc_tvs $ tcHsSigType (FunSigCtxt name1) hs_ty
           ; sigs <- mapM (instTcTySig hs_ty sigma_ty (extra_cts hs_ty) (zip wcs nwc_tvs))
                          (map unLoc names)
           ; return (sigs, nwc_tvs) }
      where
         extra_cts (L _ (HsForAllTy _ extra _ _ _)) = extra
         extra_cts _ = Nothing

    tcTySig (L loc (PatSynSig (L _ name) (_, qtvs) prov req ty))
      = setSrcSpan loc $
        do { traceTc "tcTySig {" $ ppr name $$ ppr qtvs $$ ppr prov $$ ppr req $$ ppr ty
           ; let ctxt = FunSigCtxt name
           ; tcHsTyVarBndrs qtvs $ \ qtvs' -> do
           { ty' <- tcHsSigType ctxt ty
           ; req' <- tcHsContext req
           ; prov' <- tcHsContext prov

           ; qtvs' <- mapM zonkQuantifiedTyVar qtvs'

           ; let (_, pat_ty) = tcSplitFunTys ty'
                 univ_set = tyVarsOfType pat_ty
                 (univ_tvs, ex_tvs) = partition (`elemVarSet` univ_set) qtvs'

           ; traceTc "tcTySig }" $ ppr (ex_tvs, prov') $$ ppr (univ_tvs, req') $$ ppr ty'
           ; let tpsi = TPSI{ patsig_name = name,
                              patsig_tau = ty',
                              patsig_ex = ex_tvs,
                              patsig_univ = univ_tvs,
                              patsig_prov = prov',
                              patsig_req = req' }
           ; return ([TcPatSynInfo tpsi], []) }}
    tcTySig _ = return ([], [])

    instTcTySigFromId :: Id -> TcM TcSigInfo
    instTcTySigFromId id
      = do { let loc = getSrcSpan id
           ; (tvs, theta, tau) <- tcInstType (tcInstSigTyVarsLoc loc)
                                             (idType id)
           ; return (TcSigInfo { sig_id = id, sig_loc = loc
                               , sig_tvs = [(Nothing, tv) | tv <- tvs]
                               , sig_nwcs = []
                               , sig_theta = theta, sig_tau = tau
                               , sig_extra_cts = Nothing
                               , sig_partial = False }) }

    instTcTySig :: LHsType Name -> TcType    -- HsType and corresponding TcType
                -> Maybe SrcSpan             -- Just loc <=> an extra-constraints
                                             -- wildcard is present at location loc.
                -> [(Name, TcTyVar)] -> Name -> TcM TcSigInfo
    instTcTySig hs_ty@(L loc _) sigma_ty extra_cts nwcs name
      = do { (inst_tvs, theta, tau) <- tcInstType tcInstSigTyVars sigma_ty
           ; return (TcSigInfo { sig_id  = mkLocalId name sigma_ty
                               , sig_loc = loc
                               , sig_tvs = findScopedTyVars hs_ty sigma_ty inst_tvs
                               , sig_nwcs = nwcs
                               , sig_theta = theta, sig_tau = tau
                               , sig_extra_cts = extra_cts
                               , sig_partial = isJust extra_cts || not (null nwcs) }) }

    -------------------------------
    data GeneralisationPlan
      = NoGen               -- No generalisation, no AbsBinds

      | InferGen            -- Implicit generalisation; there is an AbsBinds
           Bool             --   True <=> apply the MR; generalise only unconstrained type vars
           Bool             --   True <=> bindings mention only variables with closed types
                            --            See Note [Bindings with closed types] in TcRnTypes

      | CheckGen (LHsBind Name) TcSigInfo
                            -- One binding with a signature
                            -- Explicit generalisation; there is an AbsBinds

    -- A consequence of the no-AbsBinds choice (NoGen) is that there is
    -- no "polymorphic Id" and "monmomorphic Id"; there is just the one

    instance Outputable GeneralisationPlan where
      ppr NoGen          = ptext (sLit "NoGen")
      ppr (InferGen b c) = ptext (sLit "InferGen") <+> ppr b <+> ppr c
      ppr (CheckGen _ s) = ptext (sLit "CheckGen") <+> ppr s

    decideGeneralisationPlan
       :: DynFlags -> TcTypeEnv -> [Name]
       -> [LHsBind Name] -> TcSigFun -> GeneralisationPlan
    decideGeneralisationPlan dflags type_env bndr_names lbinds sig_fn
      | strict_pat_binds                                 = NoGen
      | Just (lbind, sig) <- one_funbind_with_sig lbinds = CheckGen lbind sig
      | mono_local_binds                                 = NoGen
      | otherwise                                        = InferGen mono_restriction closed_flag

      where
        bndr_set = mkNameSet bndr_names
        binds = map unLoc lbinds

        strict_pat_binds = any isStrictHsBind binds
           -- Strict patterns (top level bang or unboxed tuple) must not
           -- be polymorphic, because we are going to force them
           -- See Trac #4498, #8762

        mono_restriction  = xopt Opt_MonomorphismRestriction dflags
                         && any restricted binds

        is_closed_ns :: NameSet -> Bool -> Bool
        is_closed_ns ns b = foldNameSet ((&&) . is_closed_id) b ns
            -- ns are the Names referred to from the RHS of this bind

        is_closed_id :: Name -> Bool
        -- See Note [Bindings with closed types] in TcRnTypes
        is_closed_id name
          | name `elemNameSet` bndr_set
          = True              -- Ignore binders in this groups, of course
          | Just thing <- lookupNameEnv type_env name
          = case thing of
              ATcId { tct_closed = cl } -> isTopLevel cl  -- This is the key line
              ATyVar {}                 -> False          -- In-scope type variables
              AGlobal {}                -> True           --    are not closed!
              _                         -> pprPanic "is_closed_id" (ppr name)
          | otherwise
          = WARN( isInternalName name, ppr name ) True
            -- The free-var set for a top level binding mentions
            -- imported things too, so that we can report unused imports
            -- These won't be in the local type env.
            -- Ditto class method etc from the current module

        closed_flag = foldr (is_closed_ns . bind_fvs) True binds

        mono_local_binds = xopt Opt_MonoLocalBinds dflags
                        && not closed_flag

        no_sig n = noCompleteSig (sig_fn n)

        -- With OutsideIn, all nested bindings are monomorphic
        -- except a single function binding with a signature
        one_funbind_with_sig [lbind@(L _ (FunBind { fun_id = v }))]
          = case sig_fn (unLoc v) of
            Nothing -> Nothing
            Just sig | isPartialSig sig -> Nothing
            Just sig | otherwise        -> Just (lbind, sig)
        one_funbind_with_sig _
          = Nothing

        -- The Haskell 98 monomorphism resetriction
        restricted (PatBind {})                              = True
        restricted (VarBind { var_id = v })                  = no_sig v
        restricted (FunBind { fun_id = v, fun_matches = m }) = restricted_match m
                                                               && no_sig (unLoc v)
        restricted (PatSynBind {}) = panic "isRestrictedGroup/unrestricted PatSynBind"
        restricted (AbsBinds {}) = panic "isRestrictedGroup/unrestricted AbsBinds"

        restricted_match (MG { mg_alts = L _ (Match _ [] _ _) : _ }) = True
        restricted_match _                                           = False
            -- No args => like a pattern binding
            -- Some args => a function binding

    -------------------
    checkStrictBinds :: TopLevelFlag -> RecFlag
                     -> [LHsBind Name]
                     -> LHsBinds TcId -> [Id]
                     -> TcM ()
    -- Check that non-overloaded unlifted bindings are
    --      a) non-recursive,
    --      b) not top level,
    --      c) not a multiple-binding group (more or less implied by (a))

    checkStrictBinds top_lvl rec_group orig_binds tc_binds poly_ids
      | unlifted_bndrs || any_strict_pat   -- This binding group must be matched strictly
      = do  { checkTc (isNotTopLevel top_lvl)
                      (strictBindErr "Top-level" unlifted_bndrs orig_binds)
            ; checkTc (isNonRec rec_group)
                      (strictBindErr "Recursive" unlifted_bndrs orig_binds)

            ; checkTc (all is_monomorphic (bagToList tc_binds))
                      (polyBindErr orig_binds)
                -- data Ptr a = Ptr Addr#
                -- f x = let p@(Ptr y) = ... in ...
                -- Here the binding for 'p' is polymorphic, but does
                -- not mix with an unlifted binding for 'y'.  You should
                -- use a bang pattern.  Trac #6078.

            ; checkTc (isSingleton orig_binds)
                      (strictBindErr "Multiple" unlifted_bndrs orig_binds)

            -- Complain about a binding that looks lazy
            --    e.g.    let I# y = x in ...
            -- Remember, in checkStrictBinds we are going to do strict
            -- matching, so (for software engineering reasons) we insist
            -- that the strictness is manifest on each binding
            -- However, lone (unboxed) variables are ok
            ; checkTc (not any_pat_looks_lazy)
                      (unliftedMustBeBang orig_binds) }
      | otherwise
      = traceTc "csb2" (ppr poly_ids) >>
        return ()
      where
        unlifted_bndrs     = any is_unlifted poly_ids
        any_strict_pat     = any (isStrictHsBind   . unLoc) orig_binds
        any_pat_looks_lazy = any (looksLazyPatBind . unLoc) orig_binds

        is_unlifted id = case tcSplitSigmaTy (idType id) of
                           (_, _, rho) -> isUnLiftedType rho
              -- For the is_unlifted check, we need to look inside polymorphism
              -- and overloading.  E.g.  x = (# 1, True #)
              -- would get type forall a. Num a => (# a, Bool #)
              -- and we want to reject that.  See Trac #9140

        is_monomorphic (L _ (AbsBinds { abs_tvs = tvs, abs_ev_vars = evs }))
                         = null tvs && null evs
        is_monomorphic _ = True

    unliftedMustBeBang :: [LHsBind Name] -> SDoc
    unliftedMustBeBang binds
      = hang (text "Pattern bindings containing unlifted types should use an outermost bang pattern:")
           2 (vcat (map ppr binds))

    polyBindErr :: [LHsBind Name] -> SDoc
    polyBindErr binds
      = hang (ptext (sLit "You can't mix polymorphic and unlifted bindings"))
           2 (vcat [vcat (map ppr binds),
                    ptext (sLit "Probable fix: use a bang pattern")])

    strictBindErr :: String -> Bool -> [LHsBind Name] -> SDoc
    strictBindErr flavour unlifted_bndrs binds
      = hang (text flavour <+> msg <+> ptext (sLit "aren't allowed:"))
           2 (vcat (map ppr binds))
      where
        msg | unlifted_bndrs = ptext (sLit "bindings for unlifted types")
            | otherwise      = ptext (sLit "bang-pattern or unboxed-tuple bindings")

    {-
    Note [Binding scoped type variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    ************************************************************************
    *                                                                      *
    \subsection[TcBinds-errors]{Error contexts and messages}
    *                                                                      *
    ************************************************************************
    -}

    -- This one is called on LHS, when pat and grhss are both Name
    -- and on RHS, when pat is TcId and grhss is still Name
    patMonoBindsCtxt :: (OutputableBndr id, Outputable body) => LPat id -> GRHSs Name body -> SDoc
    patMonoBindsCtxt pat grhss
      = hang (ptext (sLit "In a pattern binding:")) 2 (pprPatBind pat grhss)

    typeSigCtxt :: Name -> TcSigInfo -> SDoc
    typeSigCtxt _    (TcPatSynInfo _)
      = panic "Should only be called with a TcSigInfo"
    typeSigCtxt name (TcSigInfo { sig_id = _id, sig_tvs = tvs
                                , sig_theta = theta, sig_tau = tau
                                , sig_extra_cts = extra_cts })
      = sep [ text "In" <+> pprUserTypeCtxt (FunSigCtxt name) <> colon
            , nest 2 (pprSigmaTypeExtraCts (isJust extra_cts)
                      (mkSigmaTy (map snd tvs) theta tau)) ]