Check.hs 54.1 KB
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{-
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Author: George Karachalias <george.karachalias@cs.kuleuven.be>
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Pattern Matching Coverage Checking.
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-}
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{-# LANGUAGE CPP #-}

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module Check (
        -- Actual check and pretty printing
        checkSingle, checkMatches, dsPmWarn,

        -- See Note [Type and Term Equality Propagation]
        genCaseTmCs1, genCaseTmCs2
    ) where
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#include "HsVersions.h"
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import TmOracle

import DynFlags
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import HsSyn
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import TcHsSyn
import Id
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import ConLike
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import DataCon
import Name
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import TysWiredIn
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import TyCon
import SrcLoc
import Util
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import Outputable
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import FastString
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import DsMonad    -- DsM, initTcDsForSolver, getDictsDs
import TcSimplify -- tcCheckSatisfiability
import TcType     -- toTcType, toTcTypeBag
import Bag
import ErrUtils
import MonadUtils -- MonadIO
import Var        -- EvVar
import Type
import UniqSupply
import DsGRHSs    -- isTrueLHsExpr

import Data.List     -- find
import Data.Maybe    -- isNothing, isJust, fromJust
import Control.Monad -- liftM3, forM
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{-
This module checks pattern matches for:
\begin{enumerate}
  \item Equations that are redundant
  \item Equations with inaccessible right-hand-side
  \item Exhaustiveness
\end{enumerate}
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The algorithm used is described in the paper:
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  "GADTs Meet Their Match:
     Pattern-matching Warnings That Account for GADTs, Guards, and Laziness"
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    http://people.cs.kuleuven.be/~george.karachalias/papers/p424-karachalias.pdf
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%************************************************************************
%*                                                                      *
                     Pattern Match Check Types
%*                                                                      *
%************************************************************************
-}
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type PmM a = DsM a

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data PmConstraint = TmConstraint PmExpr PmExpr -- ^ Term equalities: e ~ e
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                  | TyConstraint [EvVar]   -- ^ Type equalities
                  | BtConstraint Id        -- ^ Strictness constraints: x ~ _|_

-- The *arity* of a PatVec [p1,..,pn] is
-- the number of p1..pn that are not Guards

data PmPat p = PmCon { pm_con_con     :: DataCon
                     , pm_con_arg_tys :: [Type]
                     , pm_con_tvs     :: [TyVar]
                     , pm_con_dicts   :: [EvVar]
                     , pm_con_args    :: [p] }
               -- For PmCon arguments' meaning see @ConPatOut@ in hsSyn/HsPat.hs
             | PmVar { pm_var_id      :: Id }
             | PmLit { pm_lit_lit     :: PmLit } -- See Note [Literals in PmPat]

-- data T a where
--     MkT :: forall p q. (Eq p, Ord q) => p -> q -> T [p]
-- or  MkT :: forall p q r. (Eq p, Ord q, [p] ~ r) => p -> q -> T r

data Pattern = PmGuard PatVec PmExpr      -- ^ Guard Patterns
             | NonGuard (PmPat Pattern)   -- ^ Other Patterns

newtype ValAbs = VA (PmPat ValAbs) -- Value Abstractions

-- Not sure if this is needed
instance Outputable ValAbs where
  ppr = ppr . valAbsToPmExpr

type PatVec    = [Pattern] -- Pattern Vectors
type ValVecAbs = [ValAbs]  -- Value Vector Abstractions

data ValSetAbs   -- Reprsents a set of value vector abstractions
                 -- Notionally each value vector abstraction is a triple:
                 --   (Gamma |- us |> Delta)
                 -- where 'us'    is a ValueVec
                 --       'Delta' is a constraint
  -- INVARIANT VsaInvariant: an empty ValSetAbs is always represented by Empty
  -- INVARIANT VsaArity: the number of Cons's in any path to a leaf is the same
  -- The *arity* of a ValSetAbs is the number of Cons's in any path to a leaf
  = Empty                               -- ^ {}
  | Union ValSetAbs ValSetAbs           -- ^ S1 u S2
  | Singleton                           -- ^ { |- empty |> empty }
  | Constraint [PmConstraint] ValSetAbs -- ^ Extend Delta
  | Cons ValAbs ValSetAbs               -- ^ map (ucon u) vs

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-- | Pattern check result
--
-- * redundant clauses
-- * clauses with inaccessible RHS
-- * missing
type PmResult = ( [[LPat Id]]
                , [[LPat Id]]
                , [([PmExpr], [ComplexEq])] )
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{-
%************************************************************************
%*                                                                      *
       Entry points to the checker: checkSingle and checkMatches
%*                                                                      *
%************************************************************************
-}
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-- | Check a single pattern binding (let)
checkSingle :: Id -> Pat Id -> DsM PmResult
checkSingle var p = do
  let lp = [noLoc p]
  vec <- liftUs (translatePat p)
  vsa <- initial_uncovered [var]
  (c,d,us') <- patVectProc (vec,[]) vsa -- no guards
  us <- pruneVSA us'
  return $ case (c,d) of
    (True,  _)     -> ([],   [],   us)
    (False, True)  -> ([],   [lp], us)
    (False, False) -> ([lp], [],   us)

-- | Check a matchgroup (case, functions, etc.)
checkMatches :: [Id] -> [LMatch Id (LHsExpr Id)] -> DsM PmResult
checkMatches vars matches
  | null matches = return ([],[],[])
  | otherwise    = do
      missing    <- initial_uncovered vars
      (rs,is,us) <- go matches missing
      return (map hsLMatchPats rs, map hsLMatchPats is, us)
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  where
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    go [] missing = do
      missing' <- pruneVSA missing
      return ([], [], missing')

    go (m:ms) missing = do
      clause        <- liftUs (translateMatch m)
      (c,  d,  us ) <- patVectProc clause missing
      (rs, is, us') <- go ms us
      return $ case (c,d) of
        (True,  _)     -> (  rs,   is, us')
        (False, True)  -> (  rs, m:is, us')
        (False, False) -> (m:rs,   is, us')

-- | Generate the initial uncovered set. It initializes the
-- delta with all term and type constraints in scope.
initial_uncovered :: [Id] -> DsM ValSetAbs
initial_uncovered vars = do
  ty_cs <- TyConstraint . bagToList <$> getDictsDs
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  tm_cs <- map simpleToTmCs . bagToList <$> getTmCsDs
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  let vsa = map (VA . PmVar) vars
  return $ mkConstraint (ty_cs:tm_cs) (foldr Cons Singleton vsa)
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  where
    simpleToTmCs :: (Id, PmExpr) -> PmConstraint
    simpleToTmCs (x,e) = TmConstraint (PmExprVar x) e
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{-
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%************************************************************************
%*                                                                      *
              Transform source syntax to *our* syntax
%*                                                                      *
%************************************************************************
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-}
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-- -----------------------------------------------------------------------
-- * Utilities

nullaryConPattern :: DataCon -> Pattern
-- Nullary data constructor and nullary type constructor
nullaryConPattern con = NonGuard $
  PmCon { pm_con_con = con, pm_con_arg_tys = []
        , pm_con_tvs = [], pm_con_dicts = [], pm_con_args = [] }

truePattern :: Pattern
truePattern = nullaryConPattern trueDataCon

-- | A fake guard pattern (True <- _) used to represent cases we cannot handle
fake_pat :: Pattern
fake_pat = PmGuard [truePattern] (PmExprOther EWildPat)

vanillaConPattern :: DataCon -> [Type] -> PatVec -> Pattern
-- ADT constructor pattern => no existentials, no local constraints
vanillaConPattern con arg_tys args = NonGuard $
  PmCon { pm_con_con = con, pm_con_arg_tys = arg_tys
        , pm_con_tvs = [], pm_con_dicts = [], pm_con_args = args }

nilPattern :: Type -> Pattern
nilPattern ty = NonGuard $
  PmCon { pm_con_con = nilDataCon, pm_con_arg_tys = [ty]
        , pm_con_tvs = [], pm_con_dicts = []
        , pm_con_args = [] }

mkListPatVec :: Type -> PatVec -> PatVec -> PatVec
mkListPatVec ty xs ys = [NonGuard $ PmCon { pm_con_con = consDataCon
                                          , pm_con_arg_tys = [ty]
                                          , pm_con_tvs = [], pm_con_dicts = []
                                          , pm_con_args = xs++ys }]

mkLitPattern :: HsLit -> Pattern
mkLitPattern lit = NonGuard $ PmLit { pm_lit_lit = PmSLit lit }

-- -----------------------------------------------------------------------
-- * Transform (Pat Id) into of (PmPat Id)

translatePat :: Pat Id -> UniqSM PatVec
translatePat pat = case pat of
  WildPat ty         -> mkPatternVarsSM [ty]
  VarPat  id         -> return [idPatternVar (unLoc id)]
  ParPat p           -> translatePat (unLoc p)
  LazyPat _          -> mkPatternVarsSM [hsPatType pat] -- like a variable

  -- ignore strictness annotations for now
  BangPat p          -> translatePat (unLoc p)

  AsPat lid p -> do
     -- Note [Translating As Patterns]
    ps <- translatePat (unLoc p)
    let [e] = map valAbsToPmExpr (coercePatVec ps)
        g   = PmGuard [idPatternVar (unLoc lid)] e
    return (ps ++ [g])

  SigPatOut p _ty -> translatePat (unLoc p)

  CoPat wrapper p ty -> do
    ps      <- translatePat p
    (xp,xe) <- mkPmId2FormsSM ty
    let g = mkGuard ps (HsWrap wrapper (unLoc xe))
    return [xp,g]

  -- (n + k)  ===>   x (True <- x >= k) (n <- x-k)
  NPlusKPat (L _ n) k ge minus -> do
    (xp, xe) <- mkPmId2FormsSM (idType n)
    let ke = L (getLoc k) (HsOverLit (unLoc k))
        g1 = mkGuard [truePattern]    (OpApp xe (noLoc ge)    no_fixity ke)
        g2 = mkGuard [idPatternVar n] (OpApp xe (noLoc minus) no_fixity ke)
    return [xp, g1, g2]

  -- (fun -> pat)   ===>   x (pat <- fun x)
  ViewPat lexpr lpat arg_ty -> do
    ps <- translatePat (unLoc lpat)
    -- See Note [Guards and Approximation]
    case all cantFailPattern ps of
      True  -> do
        (xp,xe) <- mkPmId2FormsSM arg_ty
        let g = mkGuard ps (HsApp lexpr xe)
        return [xp,g]
      False -> do
        var <- mkPatternVarSM arg_ty
        return [var, fake_pat]

  -- list
  ListPat ps ty Nothing -> do
    foldr (mkListPatVec ty) [nilPattern ty] <$> translatePatVec (map unLoc ps)

  -- overloaded list
  ListPat lpats elem_ty (Just (pat_ty, _to_list))
    | Just e_ty <- splitListTyConApp_maybe pat_ty, elem_ty `eqType` e_ty ->
        -- We have to ensure that the element types are exactly the same.
        -- Otherwise, one may give an instance IsList [Int] (more specific than
        -- the default IsList [a]) with a different implementation for `toList'
        translatePat (ListPat lpats e_ty Nothing)
    | otherwise -> do
        -- See Note [Guards and Approximation]
        var <- mkPatternVarSM pat_ty
        return [var, fake_pat]

  ConPatOut { pat_con = L _ (PatSynCon _) } -> do
    -- Pattern synonyms have a "matcher"
    -- (see Note [Pattern synonym representation] in PatSyn.hs
    -- We should be able to transform (P x y)
    -- to   v (Just (x, y) <- matchP v (\x y -> Just (x,y)) Nothing
    -- That is, a combination of a variable pattern and a guard
    -- But there are complications with GADTs etc, and this isn't done yet
    var <- mkPatternVarSM (hsPatType pat)
    return [var,fake_pat]

  ConPatOut { pat_con     = L _ (RealDataCon con)
            , pat_arg_tys = arg_tys
            , pat_tvs     = ex_tvs
            , pat_dicts   = dicts
            , pat_args    = ps } -> do
    args <- translateConPatVec arg_tys ex_tvs con ps
    return [ NonGuard $ PmCon { pm_con_con     = con
                              , pm_con_arg_tys = arg_tys
                              , pm_con_tvs     = ex_tvs
                              , pm_con_dicts   = dicts
                              , pm_con_args    = args }]

  NPat (L _ ol) mb_neg _eq -> translateNPat ol mb_neg

  LitPat lit
      -- If it is a string then convert it to a list of characters
    | HsString src s <- lit ->
        foldr (mkListPatVec charTy) [nilPattern charTy] <$>
          translatePatVec (map (LitPat . HsChar src) (unpackFS s))
    | otherwise -> return [mkLitPattern lit]

  PArrPat ps ty -> do
    tidy_ps <- translatePatVec (map unLoc ps)
    let fake_con = parrFakeCon (length ps)
    return [vanillaConPattern fake_con [ty] (concat tidy_ps)]

  TuplePat ps boxity tys -> do
    tidy_ps   <- translatePatVec (map unLoc ps)
    let tuple_con = tupleDataCon boxity (length ps)
    return [vanillaConPattern tuple_con tys (concat tidy_ps)]

  -- --------------------------------------------------------------------------
  -- Not supposed to happen
  ConPatIn  {} -> panic "Check.translatePat: ConPatIn"
  SplicePat {} -> panic "Check.translatePat: SplicePat"
  SigPatIn  {} -> panic "Check.translatePat: SigPatIn"

-- | Translate an overloaded literal (see `tidyNPat' in deSugar/MatchLit.hs)
translateNPat :: HsOverLit Id -> Maybe (SyntaxExpr Id) -> UniqSM PatVec
translateNPat (OverLit val False _ ty) mb_neg
  | isStringTy ty, HsIsString src s <- val, Nothing <- mb_neg
  = translatePat (LitPat (HsString src s))
  | isIntTy    ty, HsIntegral src i <- val
  = translatePat (mk_num_lit HsInt src i)
  | isWordTy   ty, HsIntegral src i <- val
  = translatePat (mk_num_lit HsWordPrim src i)
  where
    mk_num_lit c src i = LitPat $ case mb_neg of
      Nothing -> c src i
      Just _  -> c src (-i)
translateNPat ol mb_neg
  = return [NonGuard $ PmLit { pm_lit_lit = PmOLit (isJust mb_neg) ol }]

-- | Translate a list of patterns (Note: each pattern is translated
-- to a pattern vector but we do not concatenate the results).
translatePatVec :: [Pat Id] -> UniqSM [PatVec]
translatePatVec pats = mapM translatePat pats

translateConPatVec :: [Type] -> [TyVar]
                   -> DataCon -> HsConPatDetails Id -> UniqSM PatVec
translateConPatVec _univ_tys _ex_tvs _ (PrefixCon ps)
  = concat <$> translatePatVec (map unLoc ps)
translateConPatVec _univ_tys _ex_tvs _ (InfixCon p1 p2)
  = concat <$> translatePatVec (map unLoc [p1,p2])
translateConPatVec  univ_tys  ex_tvs c (RecCon (HsRecFields fs _))
    -- Nothing matched. Make up some fresh term variables
  | null fs        = mkPatternVarsSM arg_tys
    -- The data constructor was not defined using record syntax. For the
    -- pattern to be in record syntax it should be empty (e.g. Just {}).
    -- So just like the previous case.
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  | null orig_lbls = ASSERT(null matched_lbls) mkPatternVarsSM arg_tys
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    -- Some of the fields appear, in the original order (there may be holes).
    -- Generate a simple constructor pattern and make up fresh variables for
    -- the rest of the fields
  | matched_lbls `subsetOf` orig_lbls
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  = ASSERT(length orig_lbls == length arg_tys)
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      let translateOne (lbl, ty) = case lookup lbl matched_pats of
            Just p  -> translatePat p
            Nothing -> mkPatternVarsSM [ty]
      in  concatMapM translateOne (zip orig_lbls arg_tys)
    -- The fields that appear are not in the correct order. Make up fresh
    -- variables for all fields and add guards after matching, to force the
    -- evaluation in the correct order.
  | otherwise = do
      arg_var_pats    <- mkPatternVarsSM arg_tys
      translated_pats <- forM matched_pats $ \(x,pat) -> do
        pvec <- translatePat pat
        return (x, pvec)

      let zipped = zip orig_lbls [ x | NonGuard (PmVar x) <- arg_var_pats ]
          guards = map (\(name,pvec) -> case lookup name zipped of
                            Just x  -> PmGuard pvec (PmExprVar x)
                            Nothing -> panic "translateConPatVec: lookup")
                       translated_pats

      return (arg_var_pats ++ guards)
  where
    -- The actual argument types (instantiated)
    arg_tys = dataConInstOrigArgTys c (univ_tys ++ mkTyVarTys ex_tvs)

    -- Some label information
    orig_lbls    = map flSelector $ dataConFieldLabels c
    matched_pats = [ (getName (unLoc (hsRecFieldId x)), unLoc (hsRecFieldArg x))
                   | L _ x <- fs]
    matched_lbls = [ name | (name, _pat) <- matched_pats ]

    subsetOf :: Eq a => [a] -> [a] -> Bool
    subsetOf []     _  = True
    subsetOf (_:_)  [] = False
    subsetOf (x:xs) (y:ys)
      | x == y    = subsetOf    xs  ys
      | otherwise = subsetOf (x:xs) ys

translateMatch :: LMatch Id (LHsExpr Id) -> UniqSM (PatVec,[PatVec])
translateMatch (L _ (Match _ lpats _ grhss)) = do
  pats'   <- concat <$> translatePatVec pats
  guards' <- mapM translateGuards guards
  return (pats', guards')
  where
    extractGuards :: LGRHS Id (LHsExpr Id) -> [GuardStmt Id]
    extractGuards (L _ (GRHS gs _)) = map unLoc gs

    pats   = map unLoc lpats
    guards = map extractGuards (grhssGRHSs grhss)

-- -----------------------------------------------------------------------
-- * Transform source guards (GuardStmt Id) to PmPats (Pattern)

-- | Translate a list of guard statements to a pattern vector
translateGuards :: [GuardStmt Id] -> UniqSM PatVec
translateGuards guards = do
  all_guards <- concat <$> mapM translateGuard guards
  return (replace_unhandled all_guards)
  -- It should have been (return $ all_guards) but it is too expressive.
  -- Since the term oracle does not handle all constraints we generate,
  -- we (hackily) replace all constraints the oracle cannot handle with a
  -- single one (we need to know if there is a possibility of falure).
  -- See Note [Guards and Approximation] for all guard-related approximations
  -- we implement.
  where
    replace_unhandled :: PatVec -> PatVec
    replace_unhandled gv
      | any_unhandled gv = fake_pat : [ p | p <- gv, shouldKeep p ]
      | otherwise        = gv

    any_unhandled :: PatVec -> Bool
    any_unhandled gv = any (not . shouldKeep) gv

    shouldKeep :: Pattern -> Bool
    shouldKeep (NonGuard p)
      | PmVar {} <- p      = True
      | PmCon {} <- p      = length (allConstructors (pm_con_con p)) == 1
                             && all shouldKeep (pm_con_args p)
    shouldKeep (PmGuard pv e)
      | all shouldKeep pv  = True
      | isNotPmExprOther e = True  -- expensive but we want it
    shouldKeep _other_pat  = False -- let the rest..

-- | Check whether a pattern can fail to match
cantFailPattern :: Pattern -> Bool
cantFailPattern (NonGuard p)
  | PmVar {} <- p = True
  | PmCon {} <- p = length (allConstructors (pm_con_con p)) == 1
                    && all cantFailPattern (pm_con_args p)
cantFailPattern (PmGuard pv _e)
                  = all cantFailPattern pv
cantFailPattern _ = False

-- | Translate a guard statement to Pattern
translateGuard :: GuardStmt Id -> UniqSM PatVec
translateGuard (BodyStmt   e _ _ _) = translateBoolGuard e
translateGuard (LetStmt      binds) = translateLet (unLoc binds)
translateGuard (BindStmt   p e _ _) = translateBind p e
translateGuard (LastStmt        {}) = panic "translateGuard LastStmt"
translateGuard (ParStmt         {}) = panic "translateGuard ParStmt"
translateGuard (TransStmt       {}) = panic "translateGuard TransStmt"
translateGuard (RecStmt         {}) = panic "translateGuard RecStmt"
translateGuard (ApplicativeStmt {}) = panic "translateGuard ApplicativeLastStmt"

-- | Translate let-bindings
translateLet :: HsLocalBinds Id -> UniqSM PatVec
translateLet _binds = return []

-- | Translate a pattern guard
translateBind :: LPat Id -> LHsExpr Id -> UniqSM PatVec
translateBind (L _ p) e = do
  ps <- translatePat p
  return [mkGuard ps (unLoc e)]

-- | Translate a boolean guard
translateBoolGuard :: LHsExpr Id -> UniqSM PatVec
translateBoolGuard e
  | isJust (isTrueLHsExpr e) = return []
    -- The formal thing to do would be to generate (True <- True)
    -- but it is trivial to solve so instead we give back an empty
    -- PatVec for efficiency
  | otherwise = return [mkGuard [truePattern] (unLoc e)]

{- Note [Guards and Approximation]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Even if the algorithm is really expressive, the term oracle we use is not.
Hence, several features are not translated *properly* but we approximate.
The list includes:

1. View Patterns
----------------
A view pattern @(f -> p)@ should be translated to @x (p <- f x)@. The term
oracle does not handle function applications so we know that the generated
constraints will not be handled at the end. Hence, we distinguish between two
cases:
  a) Pattern @p@ cannot fail. Then this is just a binding and we do the *right
     thing*.
  b) Pattern @p@ can fail. This means that when checking the guard, we will
     generate several cases, with no useful information. E.g.:

       h (f -> [a,b]) = ...
       h x ([a,b] <- f x) = ...

       uncovered set = { [x |> { False ~ (f x ~ [])            }]
                       , [x |> { False ~ (f x ~ (t1:[]))       }]
                       , [x |> { False ~ (f x ~ (t1:t2:t3:t4)) }] }

     So we have two problems:
       1) Since we do not print the constraints in the general case (they may
          be too many), the warning will look like this:

            Pattern match(es) are non-exhaustive
            In an equation for `h':
                Patterns not matched:
                    _
                    _
                    _
          Which is not short and not more useful than a single underscore.
       2) The size of the uncovered set increases a lot, without gaining more
          expressivity in our warnings.

     Hence, in this case, we replace the guard @([a,b] <- f x)@ with a *dummy*
     @fake_pat@: @True <- _@. That is, we record that there is a possibility
     of failure but we minimize it to a True/False. This generates a single
     warning and much smaller uncovered sets.

2. Overloaded Lists
-------------------
An overloaded list @[...]@ should be translated to @x ([...] <- toList x)@. The
problem is exactly like above, as its solution. For future reference, the code
below is the *right thing to do*:

   ListPat lpats elem_ty (Just (pat_ty, to_list))
     otherwise -> do
       (xp, xe) <- mkPmId2FormsSM pat_ty
       ps       <- translatePatVec (map unLoc lpats)
       let pats = foldr (mkListPatVec elem_ty) [nilPattern elem_ty] ps
           g    = mkGuard pats (HsApp (noLoc to_list) xe)
       return [xp,g]

3. Overloaded Literals
----------------------
The case with literals is a bit different. a literal @l@ should be translated
to @x (True <- x == from l)@. Since we want to have better warnings for
overloaded literals as it is a very common feature, we treat them differently.
They are mainly covered in Note [Undecidable Equality on Overloaded Literals].

4. N+K Patterns & Pattern Synonyms
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
An n+k pattern (n+k) should be translated to @x (True <- x >= k) (n <- x-k)@.
Since the only pattern of the three that causes failure is guard @(n <- x-k)@,
and has two possible outcomes. Hence, there is no benefit in using a dummy and
we implement the proper thing. Pattern synonyms are simply not implemented yet.
Hence, to be conservative, we generate a dummy pattern, assuming that the
pattern can fail.

5. Actual Guards
----------------
During translation, boolean guards and pattern guards are translated properly.
Let bindings though are omitted by function @translateLet@. Since they are lazy
bindings, we do not actually want to generate a (strict) equality (like we do
in the pattern bind case). Hence, we safely drop them.

Additionally, top-level guard translation (performed by @translateGuards@)
replaces guards that cannot be reasoned about (like the ones we described in
1-4) with a single @fake_pat@ to record the possibility of failure to match.

%************************************************************************
%*                                                                      *
                    Main Pattern Matching Check
%*                                                                      *
%************************************************************************
-}
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-- ----------------------------------------------------------------------------
-- * Process a vector

-- Covered, Uncovered, Divergent
process_guards :: UniqSupply -> [PatVec] -> (ValSetAbs, ValSetAbs, ValSetAbs)
process_guards _us [] = (Singleton, Empty, Empty) -- No guard == True guard
process_guards us  gs
  -- If we have a list of guards but one of them is empty (True by default)
  -- then we know that it is exhaustive (just a shortcut)
  | any null gs = (Singleton, Empty, Singleton)
  | otherwise   = go us Singleton gs
  where
    go _usupply missing []       = (Empty, missing, Empty)
    go  usupply missing (gv:gvs) = (mkUnion cs css, uss, mkUnion ds dss)
      where
        (us1, us2, us3, us4) = splitUniqSupply4 usupply

        cs = covered   us1 Singleton gv missing
        us = uncovered us2 Empty     gv missing
        ds = divergent us3 Empty     gv missing

        (css, uss, dss) = go us4 us gvs

-- ----------------------------------------------------------------------------
-- * Basic utilities

patternType :: Pattern -> Type
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patternType (PmGuard pv _) = ASSERT(patVecArity pv == 1) (patternType p)
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  where Just p = find ((==1) . patternArity) pv
patternType (NonGuard pat) = pmPatType pat

-- | Get the type out of a PmPat. For guard patterns (ps <- e) we use the type
-- of the first (or the single -WHEREVER IT IS- valid to use?) pattern
pmPatType :: PmPat p -> Type
pmPatType (PmCon { pm_con_con = con, pm_con_arg_tys = tys })
                                     = mkTyConApp (dataConTyCon con) tys
pmPatType (PmVar { pm_var_id  = x }) = idType x
pmPatType (PmLit { pm_lit_lit = l }) = pmLitType l

mkOneConFull :: Id -> UniqSupply -> DataCon -> (PmPat ValAbs, [PmConstraint])
--  *  x :: T tys, where T is an algebraic data type
--     NB: in the case of a data familiy, T is the *representation* TyCon
--     e.g.   data instance T (a,b) = T1 a b
--       leads to
--            data TPair a b = T1 a b  -- The "representation" type
--       It is TPair, not T, that is given to mkOneConFull
--
--  * 'con' K is a constructor of data type T
--
-- After instantiating the universal tyvars of K we get
--          K tys :: forall bs. Q => s1 .. sn -> T tys
--
-- Results: ValAbs:          K (y1::s1) .. (yn::sn)
--          [PmConstraint]:  Q, x ~ K y1..yn
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mkOneConFull x usupply con = (con_abs, constraints)
  where
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    (usupply1, usupply2, usupply3) = splitUniqSupply3 usupply

    res_ty = idType x -- res_ty == TyConApp (dataConTyCon cabs_con) cabs_arg_tys
    (univ_tvs, ex_tvs, eq_spec, thetas, arg_tys, _) = dataConFullSig con
    data_tc = dataConTyCon con   -- The representation TyCon
    tc_args = case splitTyConApp_maybe res_ty of
                 Just (tc, tys) -> ASSERT( tc == data_tc ) tys
                 Nothing -> pprPanic "mkOneConFull: Not TyConApp:" (ppr res_ty)

    subst1  = zipTopTvSubst univ_tvs tc_args

    (subst, ex_tvs') = cloneTyVarBndrs subst1 ex_tvs usupply1

    -- Fresh term variables (VAs) as arguments to the constructor
    arguments  = mkConVars usupply2 (substTys subst arg_tys)
    -- All constraints bound by the constructor (alpha-renamed)
    theta_cs   = substTheta subst (eqSpecPreds eq_spec ++ thetas)
    evvars     = zipWith (nameType "pm") (listSplitUniqSupply usupply3) theta_cs
    con_abs    = PmCon { pm_con_con     = con
                       , pm_con_arg_tys = tc_args
                       , pm_con_tvs     = ex_tvs'
                       , pm_con_dicts   = evvars
                       , pm_con_args    = arguments }

    constraints -- term and type constraints
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      | null evvars = [ TmConstraint (PmExprVar x) (pmPatToPmExpr con_abs) ]
      | otherwise   = [ TmConstraint (PmExprVar x) (pmPatToPmExpr con_abs)
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                      , TyConstraint evvars ]

mkConVars :: UniqSupply -> [Type] -> [ValAbs] -- ys, fresh with the given type
mkConVars us tys = zipWith mkValAbsVar (listSplitUniqSupply us) tys

tailVSA :: ValSetAbs -> ValSetAbs
tailVSA Empty               = Empty
tailVSA Singleton           = panic "tailVSA: Singleton"
tailVSA (Union vsa1 vsa2)   = tailVSA vsa1 `mkUnion` tailVSA vsa2
tailVSA (Constraint cs vsa) = cs `mkConstraint` tailVSA vsa
tailVSA (Cons _ vsa)        = vsa -- actual work

wrapK :: DataCon -> [Type] -> [TyVar] -> [EvVar] -> ValSetAbs -> ValSetAbs
wrapK con arg_tys ex_tvs dicts = go (dataConSourceArity con) emptylist
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  where
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    go :: Int -> DList ValAbs -> ValSetAbs -> ValSetAbs
    go _ _    Empty = Empty
    go 0 args vsa   = VA (PmCon { pm_con_con  = con, pm_con_arg_tys = arg_tys
                                , pm_con_tvs  = ex_tvs, pm_con_dicts = dicts
                                , pm_con_args = toList args }) `mkCons` vsa
    go _ _    Singleton           = panic "wrapK: Singleton"
    go n args (Cons vs vsa)       = go (n-1) (args `snoc` vs) vsa
    go n args (Constraint cs vsa) = cs `mkConstraint` go n args vsa
    go n args (Union vsa1 vsa2)   = go n args vsa1 `mkUnion` go n args vsa2

newtype DList a = DL { unDL :: [a] -> [a] }

toList :: DList a -> [a]
toList = ($[]) . unDL
{-# INLINE toList #-}

emptylist :: DList a
emptylist = DL id
{-# INLINE emptylist #-}

infixl `snoc`
snoc :: DList a -> a -> DList a
snoc xs x = DL (unDL xs . (x:))
{-# INLINE snoc #-}

-- ----------------------------------------------------------------------------
-- * Smart Value Set Abstraction Constructors
-- (NB: An empty value set can only be represented as `Empty')

-- | The smart constructor for Constraint (maintains VsaInvariant)
mkConstraint :: [PmConstraint] -> ValSetAbs -> ValSetAbs
mkConstraint _cs Empty                = Empty
mkConstraint cs1 (Constraint cs2 vsa) = Constraint (cs1++cs2) vsa
mkConstraint cs  other_vsa            = Constraint cs other_vsa

-- | The smart constructor for Union (maintains VsaInvariant)
mkUnion :: ValSetAbs -> ValSetAbs -> ValSetAbs
mkUnion Empty vsa = vsa
mkUnion vsa Empty = vsa
mkUnion vsa1 vsa2 = Union vsa1 vsa2

-- | The smart constructor for Cons (maintains VsaInvariant)
mkCons :: ValAbs -> ValSetAbs -> ValSetAbs
mkCons _ Empty = Empty
mkCons va vsa  = Cons va vsa

-- ----------------------------------------------------------------------------
-- * More smart constructors and fresh variable generation

mkGuard :: PatVec -> HsExpr Id -> Pattern
mkGuard pv e = PmGuard pv (hsExprToPmExpr e)

mkPmVar :: UniqSupply -> Type -> PmPat p
mkPmVar usupply ty = PmVar (mkPmId usupply ty)

idPatternVar :: Id -> Pattern
idPatternVar x = NonGuard (PmVar x)

mkPatternVar :: UniqSupply -> Type -> Pattern
mkPatternVar us ty = NonGuard (mkPmVar us ty)

mkValAbsVar :: UniqSupply -> Type -> ValAbs
mkValAbsVar us ty = VA (mkPmVar us ty)

mkPatternVarSM :: Type -> UniqSM Pattern
mkPatternVarSM ty = flip mkPatternVar ty <$> getUniqueSupplyM

mkPatternVarsSM :: [Type] -> UniqSM PatVec
mkPatternVarsSM tys = mapM mkPatternVarSM tys

mkPmId :: UniqSupply -> Type -> Id
mkPmId usupply ty = mkLocalId name ty
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  where
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    unique  = uniqFromSupply usupply
    occname = mkVarOccFS (fsLit (show unique))
    name    = mkInternalName unique occname noSrcSpan

mkPmId2FormsSM :: Type -> UniqSM (Pattern, LHsExpr Id)
mkPmId2FormsSM ty = do
  us <- getUniqueSupplyM
  let x = mkPmId us ty
  return (idPatternVar x, noLoc (HsVar (noLoc x)))

-- ----------------------------------------------------------------------------
-- * Converting between Value Abstractions, Patterns and PmExpr

valAbsToPmExpr :: ValAbs -> PmExpr
valAbsToPmExpr (VA va) = pmPatToPmExpr va

pmPatToPmExpr :: PmPat ValAbs -> PmExpr
pmPatToPmExpr (PmCon { pm_con_con  = c
                     , pm_con_args = ps }) = PmExprCon c (map valAbsToPmExpr ps)
pmPatToPmExpr (PmVar { pm_var_id   = x  }) = PmExprVar x
pmPatToPmExpr (PmLit { pm_lit_lit  = l  }) = PmExprLit l

-- Convert a pattern vector to a value list abstraction by dropping the guards
-- recursively (See Note [Translating As Patterns])
coercePatVec :: PatVec -> [ValAbs]
coercePatVec pv = [ VA (coercePmPat p) | NonGuard p <- pv]

coercePmPat :: PmPat Pattern -> PmPat ValAbs
coercePmPat (PmVar { pm_var_id  = x }) = PmVar { pm_var_id  = x }
coercePmPat (PmLit { pm_lit_lit = l }) = PmLit { pm_lit_lit = l }
coercePmPat (PmCon { pm_con_con = con, pm_con_arg_tys = arg_tys
                   , pm_con_tvs = tvs, pm_con_dicts = dicts
                   , pm_con_args = args })
  = PmCon { pm_con_con  = con, pm_con_arg_tys = arg_tys
          , pm_con_tvs  = tvs, pm_con_dicts = dicts
          , pm_con_args = coercePatVec args }

no_fixity :: a -- TODO: Can we retrieve the fixity from the operator name?
no_fixity = panic "Check: no fixity"

-- Get all constructors in the family (including given)
allConstructors :: DataCon -> [DataCon]
allConstructors = tyConDataCons . dataConTyCon

-- -----------------------------------------------------------------------
-- * Types and constraints

newEvVar :: Name -> Type -> EvVar
newEvVar name ty = mkLocalId name (toTcType ty)

nameType :: String -> UniqSupply -> Type -> EvVar
nameType name usupply ty = newEvVar idname ty
  where
    unique  = uniqFromSupply usupply
    occname = mkVarOccFS (fsLit (name++"_"++show unique))
    idname  = mkInternalName unique occname noSrcSpan

-- | Partition the constraints to type cs, term cs and forced variables
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splitConstraints [] = ([],[],Nothing)
splitConstraints (c : rest)
  = case c of
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      TyConstraint cs    -> (cs ++ ty_cs, tm_cs, bot_cs)
      TmConstraint e1 e2 -> (ty_cs, (e1,e2):tm_cs, bot_cs)
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      BtConstraint cs    -> ASSERT(isNothing bot_cs) -- NB: Only one x ~ _|_
                                  (ty_cs, tm_cs, Just cs)
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  where
    (ty_cs, tm_cs, bot_cs) = splitConstraints rest
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{-
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%************************************************************************
%*                                                                      *
                              The oracles
%*                                                                      *
%************************************************************************
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-}
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-- | Check whether at least a path in a value set
-- abstraction has satisfiable constraints.
anySatVSA :: ValSetAbs -> PmM Bool
anySatVSA vsa = notNull <$> pruneVSABound 1 vsa

pruneVSA :: ValSetAbs -> PmM [([PmExpr], [ComplexEq])]
-- Prune a Value Set abstraction, keeping only as many as we are going to print
-- plus one more. We need the additional one to be able to print "..." when the
-- uncovered are too many.
pruneVSA vsa = pruneVSABound (maximum_output+1) vsa

-- | Apply a term substitution to a value vector abstraction. All VAs are
-- transformed to PmExpr (used only before pretty printing).
substInValVecAbs :: PmVarEnv -> ValVecAbs -> [PmExpr]
substInValVecAbs subst = map (exprDeepLookup subst . valAbsToPmExpr)

mergeBotCs :: Maybe Id -> Maybe Id -> Maybe Id
mergeBotCs (Just x) Nothing  = Just x
mergeBotCs Nothing  (Just y) = Just y
mergeBotCs Nothing  Nothing  = Nothing
mergeBotCs (Just _) (Just _) = panic "mergeBotCs: two (x ~ _|_) constraints"

-- | Wrap up the term oracle's state once solving is complete. Drop any
-- information about unhandled constraints (involving HsExprs) and flatten
-- (height 1) the substitution.
wrapUpTmState :: TmState -> ([ComplexEq], PmVarEnv)
wrapUpTmState (residual, (_, subst)) = (residual, flattenPmVarEnv subst)

-- | Prune all paths in a value set abstraction with inconsistent constraints.
-- Returns only `n' value vector abstractions, when `n' is given as an argument.
pruneVSABound :: Int -> ValSetAbs -> PmM [([PmExpr], [ComplexEq])]
pruneVSABound n v = go n init_cs emptylist v
  where
    init_cs :: ([EvVar], TmState, Maybe Id)
    init_cs = ([], initialTmState, Nothing)

    go :: Int -> ([EvVar], TmState, Maybe Id) -> DList ValAbs
       -> ValSetAbs -> PmM [([PmExpr], [ComplexEq])]
    go n all_cs@(ty_cs, tm_env, bot_ct) vec in_vsa
      | n <= 0    = return [] -- no need to keep going
      | otherwise = case in_vsa of
          Empty -> return []
          Union vsa1 vsa2 -> do
            vecs1 <- go n                  all_cs vec vsa1
            vecs2 <- go (n - length vecs1) all_cs vec vsa2
            return (vecs1 ++ vecs2)
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          Singleton -> do
            -- TODO: Provide an incremental interface for the type oracle
            sat <- tyOracle (listToBag ty_cs)
            return $ case sat of
              True  -> let (residual_eqs, subst) = wrapUpTmState tm_env
                           vector = substInValVecAbs subst (toList vec)
                       in  [(vector, residual_eqs)]
              False -> []
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          Constraint cs vsa -> case splitConstraints cs of
            (new_ty_cs, new_tm_cs, new_bot_ct) ->
              case tmOracle tm_env new_tm_cs of
                Just new_tm_env ->
                  let bot = mergeBotCs new_bot_ct bot_ct
                      ans = case bot of
                              Nothing -> True                    -- covered
                              Just b  -> canDiverge b new_tm_env -- divergent
                  in  case ans of
                        True  -> go n (new_ty_cs++ty_cs,new_tm_env,bot) vec vsa
                        False -> return []
                Nothing -> return []
          Cons va vsa -> go n all_cs (vec `snoc` va) vsa

-- | This variable shows the maximum number of lines of output generated for
-- warnings. It will limit the number of patterns/equations displayed to
-- maximum_output. (TODO: add command-line option?)
maximum_output :: Int
maximum_output = 4

-- | Check whether a set of type constraints is satisfiable.
tyOracle :: Bag EvVar -> PmM Bool
tyOracle evs
  = do { ((_warns, errs), res) <- initTcDsForSolver $ tcCheckSatisfiability evs
       ; case res of
            Just sat -> return sat
            Nothing  -> pprPanic "tyOracle" (vcat $ pprErrMsgBagWithLoc errs) }
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{-
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%************************************************************************
%*                                                                      *
                             Sanity Checks
%*                                                                      *
%************************************************************************
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-}
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type PmArity = Int
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patVecArity :: PatVec -> PmArity
patVecArity = sum . map patternArity
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patternArity :: Pattern -> PmArity
patternArity (PmGuard  {}) = 0
patternArity (NonGuard {}) = 1
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{-
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%************************************************************************
%*                                                                      *
            Heart of the algorithm: Function patVectProc
%*                                                                      *
%************************************************************************
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-}
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-- | Process a single vector
patVectProc :: (PatVec, [PatVec]) -> ValSetAbs -> PmM (Bool, Bool, ValSetAbs)
patVectProc (vec,gvs) vsa = do
  us <- getUniqueSupplyM
  let (c_def, u_def, d_def) = process_guards us gvs -- default (continuation)
  (usC, usU, usD) <- getUniqueSupplyM3
  mb_c <- anySatVSA (covered   usC c_def vec vsa)
  mb_d <- anySatVSA (divergent usD d_def vec vsa)
  let vsa' = uncovered usU u_def vec vsa
  return (mb_c, mb_d, vsa')

-- | Covered, Uncovered, Divergent
covered, uncovered, divergent :: UniqSupply -> ValSetAbs
                              -> PatVec -> ValSetAbs -> ValSetAbs
covered   us gvsa vec vsa = pmTraverse us gvsa cMatcher vec vsa
uncovered us gvsa vec vsa = pmTraverse us gvsa uMatcher vec vsa
divergent us gvsa vec vsa = pmTraverse us gvsa dMatcher vec vsa

-- ----------------------------------------------------------------------------
-- * Generic traversal function
--
-- | Because we represent Value Set Abstractions as a different datatype, more
-- cases than the ones described in the paper appear. Since they are the same
-- for all three functions (covered, uncovered, divergent), function
-- `pmTraverse' handles these cases (`pmTraverse' also takes care of the
-- Guard-Case since it is common for all). The actual work is done by functions
-- `cMatcher', `uMatcher' and `dMatcher' below.

pmTraverse :: UniqSupply
           -> ValSetAbs -- gvsa
           -> PmMatcher -- what to do
           -> PatVec
           -> ValSetAbs
           -> ValSetAbs
pmTraverse _us _gvsa _rec _vec Empty      = Empty
pmTraverse _us  gvsa _rec []   Singleton  = gvsa
pmTraverse _us _gvsa _rec []   (Cons _ _) = panic "pmTraverse: cons"
pmTraverse  us  gvsa  rec vec  (Union vsa1 vsa2)
  = mkUnion (pmTraverse us1 gvsa rec vec vsa1)
            (pmTraverse us2 gvsa rec vec vsa2)
  where (us1, us2) = splitUniqSupply us
pmTraverse us gvsa rec vec (Constraint cs vsa)
  = mkConstraint cs (pmTraverse us gvsa rec vec vsa)
pmTraverse us gvsa rec (p:ps) vsa =
  case p of
    -- Guard Case
    PmGuard pv e ->
      let (us1, us2) = splitUniqSupply us
          y  = mkPmId us1 (patternType p)
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          cs = [TmConstraint (PmExprVar y) e]
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      in  mkConstraint cs $ tailVSA $
            pmTraverse us2 gvsa rec (pv++ps) (VA (PmVar y) `mkCons` vsa)

    -- Constructor/Variable/Literal Case
    NonGuard pat
      | Cons (VA va) vsa <- vsa -> rec us gvsa pat ps va vsa
      | otherwise -> panic "pmTraverse: singleton" -- can't be anything else

type PmMatcher =  UniqSupply
               -> ValSetAbs
               -> PmPat Pattern -> PatVec    -- Vector head and tail
               -> PmPat ValAbs  -> ValSetAbs -- VSA    head and tail
               -> ValSetAbs

cMatcher, uMatcher, dMatcher :: PmMatcher

-- cMatcher
-- ----------------------------------------------------------------------------

-- CVar
cMatcher us gvsa (PmVar x) ps va vsa
  = VA va `mkCons` (cs `mkConstraint` covered us gvsa ps vsa)
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  where cs = [TmConstraint (PmExprVar x) (pmPatToPmExpr va)]
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-- CLitCon
cMatcher us gvsa (PmLit l) ps (va@(PmCon {})) vsa
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  = VA va `mkCons` (cs `mkConstraint` covered us gvsa ps vsa)
  where cs = [ TmConstraint (PmExprLit l) (pmPatToPmExpr va) ]
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-- CConLit
cMatcher us gvsa (p@(PmCon {})) ps (PmLit l) vsa
  = cMatcher us3 gvsa p ps con_abs (mkConstraint cs vsa)
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  where
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    (us1, us2, us3)   = splitUniqSupply3 us
    y                 = mkPmId us1 (pmPatType p)
    (con_abs, all_cs) = mkOneConFull y us2 (pm_con_con p)
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    cs = TmConstraint (PmExprVar y) (PmExprLit l) : all_cs
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-- CConCon
cMatcher us gvsa (p@(PmCon { pm_con_con = c1, pm_con_args = args1 })) ps
                    (PmCon { pm_con_con = c2, pm_con_args = args2 }) vsa
  | c1 /= c2  = Empty
  | otherwise = wrapK c1 (pm_con_arg_tys p)
                         (pm_con_tvs     p)
                         (pm_con_dicts   p)
                         (covered us gvsa (args1 ++ ps)
                                          (foldr mkCons vsa args2))

-- CLitLit
cMatcher us gvsa (PmLit l1) ps (va@(PmLit l2)) vsa = case eqPmLit l1 l2 of
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  True  -> VA va `mkCons` covered us gvsa ps vsa -- match
  False -> Empty                                 -- mismatch
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-- CConVar
cMatcher us gvsa (p@(PmCon { pm_con_con = con })) ps (PmVar x) vsa
  = cMatcher us2 gvsa p ps con_abs (mkConstraint all_cs vsa)
  where
    (us1, us2)        = splitUniqSupply us
    (con_abs, all_cs) = mkOneConFull x us1 con
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-- CLitVar
cMatcher us gvsa (p@(PmLit l)) ps (PmVar x) vsa
  = cMatcher us gvsa p ps lit_abs (mkConstraint cs vsa)
  where
    lit_abs = PmLit l
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    cs      = [TmConstraint (PmExprVar x) (PmExprLit l)]
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-- uMatcher
-- ----------------------------------------------------------------------------
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-- UVar
uMatcher us gvsa (PmVar x) ps va vsa
  = VA va `mkCons` (cs `mkConstraint` uncovered us gvsa ps vsa)
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  where cs = [TmConstraint (PmExprVar x) (pmPatToPmExpr va)]
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-- ULitCon
uMatcher us gvsa (PmLit l) ps (va@(PmCon {})) vsa
  = uMatcher us2 gvsa (PmVar y) ps va (mkConstraint cs vsa)
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  where
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    (us1, us2) = splitUniqSupply us
    y  = mkPmId us1 (pmPatType va)
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    cs = [TmConstraint (PmExprVar y) (PmExprLit l)]
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-- UConLit
uMatcher us gvsa (p@(PmCon {})) ps (PmLit l) vsa
  = uMatcher us2 gvsa p ps (PmVar y) (mkConstraint cs vsa)
  where
    (us1, us2) = splitUniqSupply us
    y  = mkPmId us1 (pmPatType p)
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    cs = [TmConstraint (PmExprVar y) (PmExprLit l)]
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-- UConCon
uMatcher us gvsa ( p@(PmCon { pm_con_con = c1, pm_con_args = args1 })) ps
                 (va@(PmCon { pm_con_con = c2, pm_con_args = args2 })) vsa
  | c1 /= c2  = VA va `mkCons` vsa
  | otherwise = wrapK c1 (pm_con_arg_tys p)
                         (pm_con_tvs     p)
                         (pm_con_dicts   p)
                         (uncovered us gvsa (args1 ++ ps)
                                            (foldr mkCons vsa args2))

-- ULitLit
uMatcher us gvsa (PmLit l1) ps (va@(PmLit l2)) vsa = case eqPmLit l1 l2 of
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  True  -> VA va `mkCons` uncovered us gvsa ps vsa -- match
  False -> VA va `mkCons` vsa                      -- mismatch
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-- UConVar
uMatcher us gvsa (p@(PmCon { pm_con_con = con })) ps (PmVar x) vsa
  = uncovered us2 gvsa (NonGuard p : ps) inst_vsa
  where
    (us1, us2) = splitUniqSupply us

    -- Unfold the variable to all possible constructor patterns
    cons_cs  = zipWith (mkOneConFull x) (listSplitUniqSupply us1)
                                        (allConstructors con)
    add_one (va,cs) valset = mkUnion valset
                                     (mkCons (VA va) (mkConstraint cs vsa))
    inst_vsa = foldr add_one Empty cons_cs -- instantiated vsa [x mapsto K_j ys]

-- ULitVar
uMatcher us gvsa (p@(PmLit l)) ps (PmVar x) vsa
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  = mkUnion (uMatcher us gvsa p ps (PmLit l) (mkConstraint match_cs vsa))
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            (non_match_cs `mkConstraint` (VA (PmVar x) `mkCons` vsa))
  where
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    match_cs     = [ TmConstraint (PmExprVar x) (PmExprLit l)]
    non_match_cs = [ TmConstraint falsePmExpr
                                  (PmExprEq (PmExprVar x) (PmExprLit l)) ]
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-- dMatcher
-- ----------------------------------------------------------------------------

-- DVar
dMatcher us gvsa (PmVar x) ps va vsa
  = VA va `mkCons` (cs `mkConstraint` divergent us gvsa ps vsa)
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  where cs = [TmConstraint (PmExprVar x) (pmPatToPmExpr va)]
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-- DLitCon
dMatcher us gvsa (PmLit l) ps (va@(PmCon {})) vsa
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  = VA va `mkCons` (cs `mkConstraint` divergent us gvsa ps vsa)
  where cs = [ TmConstraint (PmExprLit l) (pmPatToPmExpr va) ]
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-- DConLit
dMatcher us gvsa (p@(PmCon { pm_con_con = con })) ps (PmLit l) vsa
  = dMatcher us3 gvsa p ps con_abs (mkConstraint cs vsa)
  where
    (us1, us2, us3)   = splitUniqSupply3 us
    y                 = mkPmId us1 (pmPatType p)
    (con_abs, all_cs) = mkOneConFull y us2 con
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    cs = TmConstraint (PmExprVar y) (PmExprLit l) : all_cs
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-- DConCon
dMatcher us gvsa (p@(PmCon { pm_con_con = c1, pm_con_args = args1 })) ps
                    (PmCon { pm_con_con = c2, pm_con_args = args2 }) vsa
  | c1 /= c2  = Empty
  | otherwise = wrapK c1 (pm_con_arg_tys p)
                         (pm_con_tvs     p)
                         (pm_con_dicts   p)
                         (divergent us gvsa (args1 ++ ps)
                                            (foldr mkCons vsa args2))

-- DLitLit
dMatcher us gvsa (PmLit l1) ps (va@(PmLit l2)) vsa = case eqPmLit l1 l2 of
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  True  -> VA va `mkCons` divergent us gvsa ps vsa -- match
  False -> Empty                                   -- mismatch
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-- DConVar
dMatcher us gvsa (p@(PmCon { pm_con_con = con })) ps (PmVar x) vsa
  = mkUnion (VA (PmVar x) `mkCons` mkConstraint [BtConstraint x] vsa)
            (dMatcher us2 gvsa p ps con_abs (mkConstraint all_cs vsa))
  where
    (us1, us2)        = splitUniqSupply us
    (con_abs, all_cs) = mkOneConFull x us1 con
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-- DLitVar
dMatcher us gvsa (PmLit l) ps (PmVar x) vsa
  = mkUnion (VA (PmVar x) `mkCons` mkConstraint [BtConstraint x] vsa)
            (dMatcher us gvsa (PmLit l) ps (PmLit l) (mkConstraint cs vsa))
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  where
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    cs = [TmConstraint (PmExprVar x) (PmExprLit l)]
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-- ----------------------------------------------------------------------------
-- * Propagation of term constraints inwards when checking nested matches
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{- Note [Type and Term Equality Propagation]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When checking a match it would be great to have all type and term information
available so we can get more precise results. For this reason we have functions
`addDictsDs' and `addTmCsDs' in DsMonad that store in the environment type and
term constraints (respectively) as we go deeper.
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The type constraints we propagate inwards are collected by `collectEvVarsPats'
in HsPat.hs. This handles bug #4139 ( see example
  https://ghc.haskell.org/trac/ghc/attachment/ticket/4139/GADTbug.hs )
where this is needed.

For term equalities we do less, we just generate equalities for HsCase. For
example we accurately give 2 redundancy warnings for the marked cases:

f :: [a] -> Bool
f x = case x of

  []    -> case x of        -- brings (x ~ []) in scope
             []    -> True
             (_:_) -> False -- can't happen

  (_:_) -> case x of        -- brings (x ~ (_:_)) in scope
             (_:_) -> True
             []    -> False -- can't happen

Functions `genCaseTmCs1' and `genCaseTmCs2' are responsible for generating
these constraints.
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-}

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-- | Generate equalities when checking a case expression:
--     case x of { p1 -> e1; ... pn -> en }
-- When we go deeper to check e.g. e1 we record two equalities:
-- (x ~ y), where y is the initial uncovered when checking (p1; .. ; pn)
-- and (x ~ p1).
genCaseTmCs2 :: Maybe (LHsExpr Id) -- Scrutinee
             -> [Pat Id]           -- LHS       (should have length 1)
             -> [Id]               -- MatchVars (should have length 1)
             -> DsM (Bag SimpleEq)
genCaseTmCs2 Nothing _ _ = return emptyBag
genCaseTmCs2 (Just scr) [p] [var] = liftUs $ do
  [e] <- map valAbsToPmExpr . coercePatVec <$> translatePat p
  let scr_e = lhsExprToPmExpr scr
  return $ listToBag [(var, e), (var, scr_e)]
genCaseTmCs2 _ _ _ = panic "genCaseTmCs2: HsCase"

-- | Generate a simple equality when checking a case expression:
--     case x of { matches }
-- When checking matches we record that (x ~ y) where y is the initial
-- uncovered. All matches will have to satisfy this equality.
genCaseTmCs1 :: Maybe (LHsExpr Id) -> [Id] -> Bag SimpleEq
genCaseTmCs1 Nothing     _    = emptyBag
genCaseTmCs1 (Just scr) [var] = unitBag (var, lhsExprToPmExpr scr)
genCaseTmCs1 _ _              = panic "genCaseTmCs1: HsCase"

{- Note [Literals in PmPat]
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Instead of translating a literal to a variable accompanied with a guard, we
treat them like constructor patterns. The following example from
"./libraries/base/GHC/IO/Encoding.hs" shows why:

mkTextEncoding' :: CodingFailureMode -> String -> IO TextEncoding
mkTextEncoding' cfm enc = case [toUpper c | c <- enc, c /= '-'] of
    "UTF8"    -> return $ UTF8.mkUTF8 cfm
    "UTF16"   -> return $ UTF16.mkUTF16 cfm
    "UTF16LE" -> return $ UTF16.mkUTF16le cfm
    ...

Each clause gets translated to a list of variables with an equal number of
guards. For every guard we generate two cases (equals True/equals False) which
means that we generate 2^n cases to feed the oracle with, where n is the sum of
the length of all strings that appear in the patterns. For this particular
example this means over 2^40 cases. Instead, by representing them like with
constructor we get the following:
  1. We exploit the common prefix with our representation of VSAs
  2. We prune immediately non-reachable cases
     (e.g. False == (x == "U"), True == (x == "U"))

Note [Translating As Patterns]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Instead of translating x@p as:  x (p <- x)
we instead translate it as:     p (x <- coercePattern p)
for performance reasons. For example:

  f x@True  = 1
  f y@False = 2

Gives the following with the first translation:

  x |> {x == False, x == y, y == True}

If we use the second translation we get an empty set, independently of the
oracle. Since the pattern `p' may contain guard patterns though, it cannot be
used as an expression. That's why we call `coercePatVec' to drop the guard and
`valAbsToPmExpr' to transform the value abstraction to an expression in the
guard pattern (value abstractions are a subset of expressions). We keep the
guards in the first pattern `p' though.
-}
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{-
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%************************************************************************
%*                                                                      *
      Pretty printing of exhaustiveness/redundancy check warnings
%*                                                                      *
%************************************************************************
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-}
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dsPmWarn :: DynFlags -> DsMatchContext -> DsM PmResult -> DsM ()
dsPmWarn dflags ctx@(DsMatchContext kind loc) mPmResult
  = when (flag_i || flag_u) $ do
      (redundant, inaccessible, uncovered) <- mPmResult
      let exists_r = flag_i && notNull redundant
          exists_i = flag_i && notNull inaccessible
          exists_u = flag_u && notNull uncovered
      when exists_r $ putSrcSpanDs loc (warnDs (pprEqns  redundant    rmsg))
      when exists_i $ putSrcSpanDs loc (warnDs (pprEqns  inaccessible imsg))
      when exists_u $ putSrcSpanDs loc (warnDs (pprEqnsU uncovered))
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  where
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    flag_i = wopt Opt_WarnOverlappingPatterns dflags
    flag_u = exhaustive dflags kind

    rmsg = "are redundant"
    imsg = "have inaccessible right hand side"

    pprEqns qs text = pp_context ctx (ptext (sLit text)) $ \f ->
      vcat (map (ppr_eqn f kind) (take maximum_output qs)) $$ dots qs

    pprEqnsU qs = pp_context ctx (ptext (sLit "are non-exhaustive")) $ \_ ->
      let us = map ppr_uncovered qs
      in  hang (ptext (sLit "Patterns not matched:")) 4
               (vcat (take maximum_output us) $$ dots us)

dots :: [a] -> SDoc
dots qs | qs `lengthExceeds` maximum_output = ptext (sLit "...")
        | otherwise                         = empty

exhaustive :: DynFlags -> HsMatchContext id -> Bool
exhaustive  dflags (FunRhs {})   = wopt Opt_WarnIncompletePatterns dflags
exhaustive  dflags CaseAlt       = wopt Opt_WarnIncompletePatterns dflags
exhaustive _dflags IfAlt         = False
exhaustive  dflags LambdaExpr    = wopt Opt_WarnIncompleteUniPatterns dflags
exhaustive  dflags PatBindRhs    = wopt Opt_WarnIncompleteUniPatterns dflags
exhaustive  dflags ProcExpr      = wopt Opt_WarnIncompleteUniPatterns dflags
exhaustive  dflags RecUpd        = wopt Opt_WarnIncompletePatternsRecUpd dflags
exhaustive _dflags ThPatSplice   = False
exhaustive _dflags PatSyn        = False
exhaustive _dflags ThPatQuote    = False
exhaustive _dflags (StmtCtxt {}) = False -- Don't warn about incomplete patterns
                                       -- in list comprehensions, pattern guards
                                       -- etc. They are often *supposed* to be
                                       -- incomplete

pp_context :: DsMatchContext -> SDoc -> ((SDoc -> SDoc) -> SDoc) -> SDoc
pp_context (DsMatchContext kind _loc) msg rest_of_msg_fun
  = vcat [ptext (sLit "Pattern match(es)") <+> msg,
          sep [ ptext (sLit "In") <+> ppr_match <> char ':'
              , nest 4 (rest_of_msg_fun pref)]]
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  where
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    (ppr_match, pref)
        = case kind of
             FunRhs fun -> (pprMatchContext kind, \ pp -> ppr fun <+> pp)
             _          -> (pprMatchContext kind, \ pp -> pp)

ppr_pats :: HsMatchContext Name -> [Pat Id] -> SDoc
ppr_pats kind pats
  = sep [sep (map ppr pats), matchSeparator kind, ptext (sLit "...")]

ppr_eqn :: (SDoc -> SDoc) -> HsMatchContext Name -> [LPat Id] -> SDoc
ppr_eqn prefixF kind eqn = prefixF (ppr_pats kind (map unLoc eqn))

ppr_constraint :: (SDoc,[PmLit]) -> SDoc
ppr_constraint (var, lits) = var <+> ptext (sLit "is not one of")
                                 <+> braces (pprWithCommas ppr lits)

ppr_uncovered :: ([PmExpr], [ComplexEq]) -> SDoc
ppr_uncovered (expr_vec, complex)
  | null cs   = fsep vec -- there are no literal constraints
  | otherwise = hang (fsep vec) 4 $
                  ptext (sLit "where") <+> vcat (map ppr_constraint cs)
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  where
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    sdoc_vec = mapM pprPmExprWithParens expr_vec
    (vec,cs) = runPmPprM sdoc_vec (filterComplex complex)