
-- | Stuff for sized types that does not require modules
--   "Agda.TypeChecking.Reduce" or "Agda.TypeChecking.Constraints"
--   (which import "Agda.TypeChecking.Monad").

module Agda.TypeChecking.Monad.SizedTypes where

import qualified Data.Foldable as Fold
import Data.List.NonEmpty (NonEmpty(..))
import qualified Data.List.NonEmpty as NonEmpty
import qualified Data.Traversable as Trav

import Agda.Syntax.Common
import Agda.Syntax.Internal

import Agda.TypeChecking.Monad.Base
import Agda.TypeChecking.Monad.Builtin
import Agda.TypeChecking.Monad.State
import Agda.TypeChecking.Positivity.Occurrence

import Agda.Utils.Except ( MonadError(catchError) )
import Agda.Utils.List
import Agda.Utils.Maybe
import Agda.Utils.Monad
import Agda.Utils.Pretty
import Agda.Utils.Singleton

import Agda.Utils.Impossible

------------------------------------------------------------------------
-- * Testing for type 'Size'
------------------------------------------------------------------------

-- | Result of querying whether size variable @i@ is bounded by another
--   size.
data BoundedSize
  =  BoundedLt Term -- ^ yes @i : Size< t@
  |  BoundedNo
     deriving (Eq, Show)

-- | Check if a type is the 'primSize' type. The argument should be 'reduce'd.
class IsSizeType a where
  isSizeType :: (HasOptions m, HasBuiltins m) => a -> m (Maybe BoundedSize)

instance IsSizeType a => IsSizeType (Dom a) where
  isSizeType = isSizeType . unDom

instance IsSizeType a => IsSizeType (b,a) where
  isSizeType = isSizeType . snd

instance IsSizeType a => IsSizeType (Type' a) where
  isSizeType = isSizeType . unEl

instance IsSizeType Term where
  isSizeType v = isSizeTypeTest <*> pure v

instance IsSizeType CompareAs where
  isSizeType (AsTermsOf a) = isSizeType a
  isSizeType AsSizes       = return $ Just BoundedNo
  isSizeType AsTypes       = return Nothing

isSizeTypeTest :: (HasOptions m, HasBuiltins m) => m (Term -> Maybe BoundedSize)
isSizeTypeTest =
  flip (ifM sizedTypesOption) (return $ const Nothing) $ do
    (size, sizelt) <- getBuiltinSize
    let testType (Def d [])        | Just d == size   = Just BoundedNo
        testType (Def d [Apply v]) | Just d == sizelt = Just $ BoundedLt $ unArg v
        testType _                                    = Nothing
    return testType

getBuiltinDefName :: (HasBuiltins m) => String -> m (Maybe QName)
getBuiltinDefName s = fromDef <$> getBuiltin' s
  where
    fromDef (Just (Def d [])) = Just d
    fromDef _                 = Nothing

getBuiltinSize :: (HasBuiltins m) => m (Maybe QName, Maybe QName)
getBuiltinSize = do
  size   <- getBuiltinDefName builtinSize
  sizelt <- getBuiltinDefName builtinSizeLt
  return (size, sizelt)

isSizeNameTest :: (HasOptions m, HasBuiltins m) => m (QName -> Bool)
isSizeNameTest = ifM sizedTypesOption
  isSizeNameTestRaw
  (return $ const False)

isSizeNameTestRaw :: (HasOptions m, HasBuiltins m) => m (QName -> Bool)
isSizeNameTestRaw = do
  (size, sizelt) <- getBuiltinSize
  return $ (`elem` [size, sizelt]) . Just

-- | Test whether OPTIONS --sized-types and whether
--   the size built-ins are defined.
haveSizedTypes :: TCM Bool
haveSizedTypes = do
    Def _ [] <- primSize
    Def _ [] <- primSizeInf
    Def _ [] <- primSizeSuc
    sizedTypesOption
  `catchError` \_ -> return False

-- | Test whether the SIZELT builtin is defined.
haveSizeLt :: TCM Bool
haveSizeLt = isJust <$> getBuiltinDefName builtinSizeLt

-- | Add polarity info to a SIZE builtin.
builtinSizeHook :: String -> QName -> Type -> TCM ()
builtinSizeHook s q t = do
  when (s `elem` [builtinSizeLt, builtinSizeSuc]) $ do
    modifySignature $ updateDefinition q
      $ updateDefPolarity       (const [Covariant])
      . updateDefArgOccurrences (const [StrictPos])
  when (s == builtinSizeMax) $ do
    modifySignature $ updateDefinition q
      $ updateDefPolarity       (const [Covariant, Covariant])
      . updateDefArgOccurrences (const [StrictPos, StrictPos])
{-
      . updateDefType           (const tmax)
  where
    -- TODO: max : (i j : Size) -> Size< (suc (max i j))
    tmax =
-}

------------------------------------------------------------------------
-- * Constructors
------------------------------------------------------------------------

-- | The sort of built-in types @SIZE@ and @SIZELT@.
sizeSort :: Sort
sizeSort = mkType 0

-- | The type of built-in types @SIZE@ and @SIZELT@.
sizeUniv :: Type
sizeUniv = sort $ sizeSort

-- | The built-in type @SIZE@ with user-given name.
sizeType_ :: QName -> Type
sizeType_ size = El sizeSort $ Def size []

-- | The built-in type @SIZE@.
sizeType :: (HasBuiltins m, MonadTCEnv m, ReadTCState m) => m Type
sizeType = El sizeSort . fromMaybe __IMPOSSIBLE__ <$> getBuiltin' builtinSize

-- | The name of @SIZESUC@.
sizeSucName :: (HasBuiltins m, HasOptions m) => m (Maybe QName)
sizeSucName = do
  ifM (not <$> sizedTypesOption) (return Nothing) $ do
    getBuiltin' builtinSizeSuc >>= \case
      Just (Def x []) -> return $ Just x
      _               -> return Nothing

sizeSuc :: HasBuiltins m => Nat -> Term -> m Term
sizeSuc n v | n < 0     = __IMPOSSIBLE__
            | n == 0    = return v
            | otherwise = do
  Def suc [] <- fromMaybe __IMPOSSIBLE__ <$> getBuiltin' builtinSizeSuc
  return $ case iterate (sizeSuc_ suc) v !!! n of
             Nothing -> __IMPOSSIBLE__
             Just t  -> t

sizeSuc_ :: QName -> Term -> Term
sizeSuc_ suc v = Def suc [Apply $ defaultArg v]

-- | Transform list of terms into a term build from binary maximum.
sizeMax :: (HasBuiltins m, MonadError TCErr m, MonadTCEnv m, ReadTCState m)
        => NonEmpty Term -> m Term
sizeMax vs = case vs of
  v :| [] -> return v
  vs  -> do
    Def max [] <- primSizeMax
    return $ foldr1 (\ u v -> Def max $ map (Apply . defaultArg) [u,v]) vs


------------------------------------------------------------------------
-- * Viewing and unviewing sizes
------------------------------------------------------------------------

-- | A useful view on sizes.
data SizeView = SizeInf | SizeSuc Term | OtherSize Term

-- | Expects argument to be 'reduce'd.
sizeView :: (HasBuiltins m, MonadError TCErr m, MonadTCEnv m, ReadTCState m)
         => Term -> m SizeView
sizeView v = do
  Def inf [] <- primSizeInf
  Def suc [] <- primSizeSuc
  case v of
    Def x []        | x == inf -> return SizeInf
    Def x [Apply u] | x == suc -> return $ SizeSuc (unArg u)
    _                          -> return $ OtherSize v

type Offset = Nat

-- | A deep view on sizes.
data DeepSizeView
  = DSizeInf
  | DSizeVar Nat Offset
  | DSizeMeta MetaId Elims Offset
  | DOtherSize Term
  deriving (Show)

instance Pretty DeepSizeView where
  pretty = \case
    DSizeInf        -> "∞"
    DSizeVar n o     -> text ("@" ++ show n) <+> "+" <+> pretty o
    DSizeMeta x es o -> pretty (MetaV x es) <+> "+" <+> pretty o
    DOtherSize t     -> pretty t

data SizeViewComparable a
  = NotComparable
  | YesAbove DeepSizeView a
  | YesBelow DeepSizeView a
  deriving (Functor)

-- | @sizeViewComparable v w@ checks whether @v >= w@ (then @Left@)
--   or @v <= w@ (then @Right@).  If uncomparable, it returns @NotComparable@.
sizeViewComparable :: DeepSizeView -> DeepSizeView -> SizeViewComparable ()
sizeViewComparable v w = case (v,w) of
  (DSizeInf, _) -> YesAbove w ()
  (_, DSizeInf) -> YesBelow w ()
  (DSizeVar x n, DSizeVar y m) | x == y -> if n >= m then YesAbove w () else YesBelow w ()
  _ -> NotComparable

sizeViewSuc_ :: QName -> DeepSizeView -> DeepSizeView
sizeViewSuc_ suc v = case v of
  DSizeInf         -> DSizeInf
  DSizeVar i n     -> DSizeVar i (n + 1)
  DSizeMeta x vs n -> DSizeMeta x vs (n + 1)
  DOtherSize u     -> DOtherSize $ sizeSuc_ suc u

-- | @sizeViewPred k v@ decrements @v@ by @k@ (must be possible!).
sizeViewPred :: Nat -> DeepSizeView -> DeepSizeView
sizeViewPred 0 v = v
sizeViewPred k v = case v of
  DSizeInf -> DSizeInf
  DSizeVar  i    n | n >= k -> DSizeVar  i    (n - k)
  DSizeMeta x vs n | n >= k -> DSizeMeta x vs (n - k)
  _ -> __IMPOSSIBLE__

-- | @sizeViewOffset v@ returns the number of successors or Nothing when infty.
sizeViewOffset :: DeepSizeView -> Maybe Offset
sizeViewOffset v = case v of
  DSizeInf         -> Nothing
  DSizeVar i n     -> Just n
  DSizeMeta x vs n -> Just n
  DOtherSize u     -> Just 0

-- | Remove successors common to both sides.
removeSucs :: (DeepSizeView, DeepSizeView) -> (DeepSizeView, DeepSizeView)
removeSucs (v, w) = (sizeViewPred k v, sizeViewPred k w)
  where k = case (sizeViewOffset v, sizeViewOffset w) of
              (Just  n, Just  m) -> min n m
              (Just  n, Nothing) -> n
              (Nothing, Just  m) -> m
              (Nothing, Nothing) -> 0

-- | Turn a size view into a term.
unSizeView :: SizeView -> TCM Term
unSizeView SizeInf       = primSizeInf
unSizeView (SizeSuc v)   = sizeSuc 1 v
unSizeView (OtherSize v) = return v

unDeepSizeView :: (HasBuiltins m, MonadError TCErr m, MonadTCEnv m, ReadTCState m)
               => DeepSizeView -> m Term
unDeepSizeView v = case v of
  DSizeInf         -> primSizeInf
  DSizeVar i     n -> sizeSuc n $ var i
  DSizeMeta x us n -> sizeSuc n $ MetaV x us
  DOtherSize u     -> return u

------------------------------------------------------------------------
-- * View on sizes where maximum is pulled to the top
------------------------------------------------------------------------

type SizeMaxView = NonEmpty DeepSizeView
type SizeMaxView' = [DeepSizeView]

maxViewMax :: SizeMaxView -> SizeMaxView -> SizeMaxView
maxViewMax v w = case (v,w) of
  (DSizeInf :| _, _) -> singleton DSizeInf
  (_, DSizeInf :| _) -> singleton DSizeInf
  _                 -> Fold.foldr maxViewCons w v

-- | @maxViewCons v ws = max v ws@.  It only adds @v@ to @ws@ if it is not
--   subsumed by an element of @ws@.
maxViewCons :: DeepSizeView -> SizeMaxView -> SizeMaxView
maxViewCons _ (DSizeInf :| _) = singleton DSizeInf
maxViewCons DSizeInf _        = singleton DSizeInf
maxViewCons v ws = case sizeViewComparableWithMax v ws of
  NotComparable  -> NonEmpty.cons v ws
  YesAbove _ ws' -> v :| ws'
  YesBelow{}     -> ws

-- | @sizeViewComparableWithMax v ws@ tries to find @w@ in @ws@ that compares with @v@
--   and singles this out.
--   Precondition: @v /= DSizeInv@.
sizeViewComparableWithMax :: DeepSizeView -> SizeMaxView -> SizeViewComparable SizeMaxView'
sizeViewComparableWithMax v (w :| ws) =
  case (ws, sizeViewComparable v w) of
    (w':ws', NotComparable) -> fmap (w:) $ sizeViewComparableWithMax v (w' :| ws')
    (ws    , r)             -> fmap (const ws) r


maxViewSuc_ :: QName -> SizeMaxView -> SizeMaxView
maxViewSuc_ suc = fmap (sizeViewSuc_ suc)

unMaxView :: (HasBuiltins m, MonadError TCErr m, MonadTCEnv m, ReadTCState m)
          => SizeMaxView -> m Term
unMaxView vs = sizeMax =<< Trav.mapM unDeepSizeView vs
