    -- (c) The University of Glasgow 2012

    {-# LANGUAGE CPP, DeriveDataTypeable, GADTs, ScopedTypeVariables #-}

    -- | Module for coercion axioms, used to represent type family instances
    -- and newtypes

    module CoAxiom (
           Branched, Unbranched, BranchIndex, BranchList(..),
           toBranchList, fromBranchList,
           toBranchedList, toUnbranchedList,
           brListLength, brListNth, brListMap, brListFoldr, brListMapM,
           brListFoldlM_, brListZipWith,

           CoAxiom(..), CoAxBranch(..),

           toBranchedAxiom, toUnbranchedAxiom,
           coAxiomName, coAxiomArity, coAxiomBranches,
           coAxiomTyCon, isImplicitCoAxiom, coAxiomNumPats,
           coAxiomNthBranch, coAxiomSingleBranch_maybe, coAxiomRole,
           coAxiomSingleBranch, coAxBranchTyVars, coAxBranchRoles,
           coAxBranchLHS, coAxBranchRHS, coAxBranchSpan, coAxBranchIncomps,
           placeHolderIncomps,

           Role(..), fsFromRole,

           CoAxiomRule(..), Eqn,
           BuiltInSynFamily(..), trivialBuiltInFamily
           ) where

    import {-# SOURCE #-} TypeRep ( Type )
    import {-# SOURCE #-} TyCon ( TyCon )
    import Outputable
    import FastString
    import Name
    import Unique
    import Var
    import Util
    import Binary
    import Pair
    import BasicTypes
    import Data.Typeable ( Typeable )
    import SrcLoc
    import qualified Data.Data as Data

    #include "HsVersions.h"

    {-
    Note [Coercion axiom branches]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    In order to allow type family instance groups, an axiom needs to contain an
    ordered list of alternatives, called branches. The kind of the coercion built
    from an axiom is determined by which index is used when building the coercion
    from the axiom.

    For example, consider the axiom derived from the following declaration:

    type instance where
      F [Int] = Bool
      F [a]   = Double
      F (a b) = Char

    This will give rise to this axiom:

    axF :: {                                           F [Int] ~ Bool
           ; forall (a :: *).                          F [a]   ~ Double
           ; forall (k :: BOX) (a :: k -> *) (b :: k). F (a b) ~ Char
           }

    The axiom is used with the AxiomInstCo constructor of Coercion. If we wish
    to have a coercion showing that F (Maybe Int) ~ Char, it will look like

    axF[2] <*> <Maybe> <Int> :: F (Maybe Int) ~ Char
    -- or, written using concrete-ish syntax --
    AxiomInstCo axF 2 [Refl *, Refl Maybe, Refl Int]

    Note that the index is 0-based.

    For type-checking, it is also necessary to check that no previous pattern
    can unify with the supplied arguments. After all, it is possible that some
    of the type arguments are lambda-bound type variables whose instantiation may
    cause an earlier match among the branches. We wish to prohibit this behavior,
    so the type checker rules out the choice of a branch where a previous branch
    can unify. See also [Branched instance checking] in FamInstEnv.hs.

    For example, the following is malformed, where 'a' is a lambda-bound type
    variable:

    axF[2] <*> <a> <Bool> :: F (a Bool) ~ Char

    Why? Because a might be instantiated with [], meaning that branch 1 should
    apply, not branch 2. This is a vital consistency check; without it, we could
    derive Int ~ Bool, and that is a Bad Thing.

    Note [Branched axioms]
    ~~~~~~~~~~~~~~~~~~~~~~~
    Although a CoAxiom has the capacity to store many branches, in certain cases,
    we want only one. These cases are in data/newtype family instances, newtype
    coercions, and type family instances declared with "type instance ...", not
    "type instance where". Furthermore, these unbranched axioms are used in a
    variety of places throughout GHC, and it would difficult to generalize all of
    that code to deal with branched axioms, especially when the code can be sure
    of the fact that an axiom is indeed a singleton. At the same time, it seems
    dangerous to assume singlehood in various places through GHC.

    The solution to this is to label a CoAxiom with a phantom type variable
    declaring whether it is known to be a singleton or not. The list of branches
    is stored using a special form of list, declared below, that ensures that the
    type variable is accurate.

    As of this writing (Dec 2012), it would not be appropriate to use a promoted
    type as the phantom type, so we use empty datatypes. We wish to have GHC
    remain compilable with GHC 7.2.1. If you are revising this code and GHC no
    longer needs to remain compatible with GHC 7.2.x, then please update this
    code to use promoted types.

    ************************************************************************
    *                                                                      *
                        Branch lists
    *                                                                      *
    ************************************************************************
    -}

    type BranchIndex = Int  -- The index of the branch in the list of branches
                            -- Counting from zero

    -- the phantom type labels
    data Unbranched deriving Typeable
    data Branched deriving Typeable

    data BranchList a br where
      FirstBranch :: a -> BranchList a br
      NextBranch :: a -> BranchList a br -> BranchList a Branched

    -- convert to/from lists
    toBranchList :: [a] -> BranchList a Branched
    toBranchList [] = pprPanic "toBranchList" empty
    toBranchList [b] = FirstBranch b
    toBranchList (h:t) = NextBranch h (toBranchList t)

    fromBranchList :: BranchList a br -> [a]
    fromBranchList (FirstBranch b) = [b]
    fromBranchList (NextBranch h t) = h : (fromBranchList t)

    -- convert from any BranchList to a Branched BranchList
    toBranchedList :: BranchList a br -> BranchList a Branched
    toBranchedList (FirstBranch b) = FirstBranch b
    toBranchedList (NextBranch h t) = NextBranch h t

    -- convert from any BranchList to an Unbranched BranchList
    toUnbranchedList :: BranchList a br -> BranchList a Unbranched
    toUnbranchedList (FirstBranch b) = FirstBranch b
    toUnbranchedList _ = pprPanic "toUnbranchedList" empty

    -- length
    brListLength :: BranchList a br -> Int
    brListLength (FirstBranch _) = 1
    brListLength (NextBranch _ t) = 1 + brListLength t

    -- lookup
    brListNth :: BranchList a br -> BranchIndex -> a
    brListNth (FirstBranch b) 0 = b
    brListNth (NextBranch h _) 0 = h
    brListNth (NextBranch _ t) n = brListNth t (n-1)
    brListNth _ _ = pprPanic "brListNth" empty

    -- map, fold
    brListMap :: (a -> b) -> BranchList a br -> [b]
    brListMap f (FirstBranch b) = [f b]
    brListMap f (NextBranch h t) = f h : (brListMap f t)

    brListFoldr :: (a -> b -> b) -> b -> BranchList a br -> b
    brListFoldr f x (FirstBranch b) = f b x
    brListFoldr f x (NextBranch h t) = f h (brListFoldr f x t)

    brListMapM :: Monad m => (a -> m b) -> BranchList a br -> m [b]
    brListMapM f (FirstBranch b) = f b >>= \fb -> return [fb]
    brListMapM f (NextBranch h t) = do { fh <- f h
                                       ; ft <- brListMapM f t
                                       ; return (fh : ft) }

    brListFoldlM_ :: forall a b m br. Monad m
                  => (a -> b -> m a) -> a -> BranchList b br -> m ()
    brListFoldlM_ f z brs = do { _ <- go z brs
                               ; return () }
      where go :: forall br'. Monad m => a -> BranchList b br' -> m a
            go acc (FirstBranch b)  = f acc b
            go acc (NextBranch h t) = do { fh <- f acc h
                                         ; go fh t }

    -- zipWith
    brListZipWith :: (a -> b -> c) -> BranchList a br1 -> BranchList b br2 -> [c]
    brListZipWith f (FirstBranch a) (FirstBranch b) = [f a b]
    brListZipWith f (FirstBranch a) (NextBranch b _) = [f a b]
    brListZipWith f (NextBranch a _) (FirstBranch b) = [f a b]
    brListZipWith f (NextBranch a ta) (NextBranch b tb) = f a b : brListZipWith f ta tb

    -- pretty-printing

    instance Outputable a => Outputable (BranchList a br) where
      ppr = ppr . fromBranchList

    {-
    ************************************************************************
    *                                                                      *
                        Coercion axioms
    *                                                                      *
    ************************************************************************

    Note [Storing compatibility]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    During axiom application, we need to be aware of which branches are compatible
    with which others. The full explanation is in Note [Compatibility] in
    FamInstEnv. (The code is placed there to avoid a dependency from CoAxiom on
    the unification algorithm.) Although we could theoretically compute
    compatibility on the fly, this is silly, so we store it in a CoAxiom.

    Specifically, each branch refers to all other branches with which it is
    incompatible. This list might well be empty, and it will always be for the
    first branch of any axiom.

    CoAxBranches that do not (yet) belong to a CoAxiom should have a panic thunk
    stored in cab_incomps. The incompatibilities are properly a property of the
    axiom as a whole, and they are computed only when the final axiom is built.

    During serialization, the list is converted into a list of the indices
    of the branches.
    -}

    -- | A 'CoAxiom' is a \"coercion constructor\", i.e. a named equality axiom.

    -- If you edit this type, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in coreSyn/CoreLint.lhs
    data CoAxiom br
      = CoAxiom                   -- Type equality axiom.
        { co_ax_unique   :: Unique        -- unique identifier
        , co_ax_name     :: Name          -- name for pretty-printing
        , co_ax_role     :: Role          -- role of the axiom's equality
        , co_ax_tc       :: TyCon         -- the head of the LHS patterns
        , co_ax_branches :: BranchList CoAxBranch br
                                          -- the branches that form this axiom
        , co_ax_implicit :: Bool          -- True <=> the axiom is "implicit"
                                          -- See Note [Implicit axioms]
             -- INVARIANT: co_ax_implicit == True implies length co_ax_branches == 1.
        }
      deriving Typeable

    data CoAxBranch
      = CoAxBranch
        { cab_loc      :: SrcSpan       -- Location of the defining equation
                                        -- See Note [CoAxiom locations]
        , cab_tvs      :: [TyVar]       -- Bound type variables; not necessarily fresh
                                        -- See Note [CoAxBranch type variables]
        , cab_roles    :: [Role]        -- See Note [CoAxBranch roles]
        , cab_lhs      :: [Type]        -- Type patterns to match against
        , cab_rhs      :: Type          -- Right-hand side of the equality
        , cab_incomps  :: [CoAxBranch]  -- The previous incompatible branches
                                        -- See Note [Storing compatibility]
        }
      deriving Typeable

    toBranchedAxiom :: CoAxiom br -> CoAxiom Branched
    toBranchedAxiom (CoAxiom unique name role tc branches implicit)
      = CoAxiom unique name role tc (toBranchedList branches) implicit

    toUnbranchedAxiom :: CoAxiom br -> CoAxiom Unbranched
    toUnbranchedAxiom (CoAxiom unique name role tc branches implicit)
      = CoAxiom unique name role tc (toUnbranchedList branches) implicit

    coAxiomNumPats :: CoAxiom br -> Int
    coAxiomNumPats = length . coAxBranchLHS . (flip coAxiomNthBranch 0)

    coAxiomNthBranch :: CoAxiom br -> BranchIndex -> CoAxBranch
    coAxiomNthBranch (CoAxiom { co_ax_branches = bs }) index
      = brListNth bs index

    coAxiomArity :: CoAxiom br -> BranchIndex -> Arity
    coAxiomArity ax index
      = length $ cab_tvs $ coAxiomNthBranch ax index

    coAxiomName :: CoAxiom br -> Name
    coAxiomName = co_ax_name

    coAxiomRole :: CoAxiom br -> Role
    coAxiomRole = co_ax_role

    coAxiomBranches :: CoAxiom br -> BranchList CoAxBranch br
    coAxiomBranches = co_ax_branches

    coAxiomSingleBranch_maybe :: CoAxiom br -> Maybe CoAxBranch
    coAxiomSingleBranch_maybe (CoAxiom { co_ax_branches = branches })
      | FirstBranch br <- branches
      = Just br
      | otherwise
      = Nothing

    coAxiomSingleBranch :: CoAxiom Unbranched -> CoAxBranch
    coAxiomSingleBranch (CoAxiom { co_ax_branches = FirstBranch br }) = br

    coAxiomTyCon :: CoAxiom br -> TyCon
    coAxiomTyCon = co_ax_tc

    coAxBranchTyVars :: CoAxBranch -> [TyVar]
    coAxBranchTyVars = cab_tvs

    coAxBranchLHS :: CoAxBranch -> [Type]
    coAxBranchLHS = cab_lhs

    coAxBranchRHS :: CoAxBranch -> Type
    coAxBranchRHS = cab_rhs

    coAxBranchRoles :: CoAxBranch -> [Role]
    coAxBranchRoles = cab_roles

    coAxBranchSpan :: CoAxBranch -> SrcSpan
    coAxBranchSpan = cab_loc

    isImplicitCoAxiom :: CoAxiom br -> Bool
    isImplicitCoAxiom = co_ax_implicit

    coAxBranchIncomps :: CoAxBranch -> [CoAxBranch]
    coAxBranchIncomps = cab_incomps

    -- See Note [Compatibility checking] in FamInstEnv
    placeHolderIncomps :: [CoAxBranch]
    placeHolderIncomps = panic "placeHolderIncomps"

    {-
    Note [CoAxBranch type variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    In the case of a CoAxBranch of an associated type-family instance,
    we use the *same* type variables (where possible) as the
    enclosing class or instance.  Consider
       class C a b where
         type F x b
         type F [y] b = ...     -- Second param must be b

       instance C Int [z] where
         type F Int [z] = ...   -- Second param must be [z]

    In the CoAxBranch in the instance decl (F Int [z]) we use the
    same 'z', so that it's easy to check that that type is the same
    as that in the instance header.

    Similarly in the CoAxBranch for the default decl for F in the
    class decl, we use the same 'b' to make the same check easy.

    So, unlike FamInsts, there is no expectation that the cab_tvs
    are fresh wrt each other, or any other CoAxBranch.

    Note [CoAxBranch roles]
    ~~~~~~~~~~~~~~~~~~~~~~~
    Consider this code:

      newtype Age = MkAge Int
      newtype Wrap a = MkWrap a

      convert :: Wrap Age -> Int
      convert (MkWrap (MkAge i)) = i

    We want this to compile to:

      NTCo:Wrap :: forall a. Wrap a ~R a
      NTCo:Age  :: Age ~R Int
      convert = \x -> x |> (NTCo:Wrap[0] NTCo:Age[0])

    But, note that NTCo:Age is at role R. Thus, we need to be able to pass
    coercions at role R into axioms. However, we don't *always* want to be able to
    do this, as it would be disastrous with type families. The solution is to
    annotate the arguments to the axiom with roles, much like we annotate tycon
    tyvars. Where do these roles get set? Newtype axioms inherit their roles from
    the newtype tycon; family axioms are all at role N.

    Note [CoAxiom locations]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    The source location of a CoAxiom is stored in two places in the
    datatype tree.
      * The first is in the location info buried in the Name of the
        CoAxiom. This span includes all of the branches of a branched
        CoAxiom.
      * The second is in the cab_loc fields of the CoAxBranches.

    In the case of a single branch, we can extract the source location of
    the branch from the name of the CoAxiom. In other cases, we need an
    explicit SrcSpan to correctly store the location of the equation
    giving rise to the FamInstBranch.

    Note [Implicit axioms]
    ~~~~~~~~~~~~~~~~~~~~~~
    See also Note [Implicit TyThings] in HscTypes
    * A CoAxiom arising from data/type family instances is not "implicit".
      That is, it has its own IfaceAxiom declaration in an interface file

    * The CoAxiom arising from a newtype declaration *is* "implicit".
      That is, it does not have its own IfaceAxiom declaration in an
      interface file; instead the CoAxiom is generated by type-checking
      the newtype declaration
    -}

    instance Eq (CoAxiom br) where
        a == b = case (a `compare` b) of { EQ -> True;   _ -> False }
        a /= b = case (a `compare` b) of { EQ -> False;  _ -> True  }

    instance Ord (CoAxiom br) where
        a <= b = case (a `compare` b) of { LT -> True;  EQ -> True;  GT -> False }
        a <  b = case (a `compare` b) of { LT -> True;  EQ -> False; GT -> False }
        a >= b = case (a `compare` b) of { LT -> False; EQ -> True;  GT -> True  }
        a >  b = case (a `compare` b) of { LT -> False; EQ -> False; GT -> True  }
        compare a b = getUnique a `compare` getUnique b

    instance Uniquable (CoAxiom br) where
        getUnique = co_ax_unique

    instance Outputable (CoAxiom br) where
        ppr = ppr . getName

    instance NamedThing (CoAxiom br) where
        getName = co_ax_name

    instance Typeable br => Data.Data (CoAxiom br) where
        -- don't traverse?
        toConstr _   = abstractConstr "CoAxiom"
        gunfold _ _  = error "gunfold"
        dataTypeOf _ = mkNoRepType "CoAxiom"

    {-
    ************************************************************************
    *                                                                      *
                        Roles
    *                                                                      *
    ************************************************************************

    Roles are defined here to avoid circular dependencies.
    -}

    -- See Note [Roles] in Coercion
    -- defined here to avoid cyclic dependency with Coercion
    data Role = Nominal | Representational | Phantom
      deriving (Eq, Data.Data, Data.Typeable)

    -- These names are slurped into the parser code. Changing these strings
    -- will change the **surface syntax** that GHC accepts! If you want to
    -- change only the pretty-printing, do some replumbing. See
    -- mkRoleAnnotDecl in RdrHsSyn
    fsFromRole :: Role -> FastString
    fsFromRole Nominal          = fsLit "nominal"
    fsFromRole Representational = fsLit "representational"
    fsFromRole Phantom          = fsLit "phantom"

    instance Outputable Role where
      ppr = ftext . fsFromRole

    instance Binary Role where
      put_ bh Nominal          = putByte bh 1
      put_ bh Representational = putByte bh 2
      put_ bh Phantom          = putByte bh 3

      get bh = do tag <- getByte bh
                  case tag of 1 -> return Nominal
                              2 -> return Representational
                              3 -> return Phantom
                              _ -> panic ("get Role " ++ show tag)

    {-
    ************************************************************************
    *                                                                      *
                        CoAxiomRule
                  Rules for building Evidence
    *                                                                      *
    ************************************************************************

    Conditional axioms.  The general idea is that a `CoAxiomRule` looks like this:

        forall as. (r1 ~ r2, s1 ~ s2) => t1 ~ t2

    My intention is to reuse these for both (~) and (~#).
    The short-term plan is to use this datatype to represent the type-nat axioms.
    In the longer run, it may be good to unify this and `CoAxiom`,
    as `CoAxiom` is the special case when there are no assumptions.
    -}

    -- | A more explicit representation for `t1 ~ t2`.
    type Eqn = Pair Type

    -- | For now, we work only with nominal equality.
    data CoAxiomRule = CoAxiomRule
      { coaxrName      :: FastString
      , coaxrTypeArity :: Int       -- number of type argumentInts
      , coaxrAsmpRoles :: [Role]    -- roles of parameter equations
      , coaxrRole      :: Role      -- role of resulting equation
      , coaxrProves    :: [Type] -> [Eqn] -> Maybe Eqn
            -- ^ coaxrProves returns @Nothing@ when it doesn't like
            -- the supplied arguments.  When this happens in a coercion
            -- that means that the coercion is ill-formed, and Core Lint
            -- checks for that.
      } deriving Typeable

    instance Data.Data CoAxiomRule where
      -- don't traverse?
      toConstr _   = abstractConstr "CoAxiomRule"
      gunfold _ _  = error "gunfold"
      dataTypeOf _ = mkNoRepType "CoAxiomRule"

    instance Uniquable CoAxiomRule where
      getUnique = getUnique . coaxrName

    instance Eq CoAxiomRule where
      x == y = coaxrName x == coaxrName y

    instance Ord CoAxiomRule where
      compare x y = compare (coaxrName x) (coaxrName y)

    instance Outputable CoAxiomRule where
      ppr = ppr . coaxrName

    -- Type checking of built-in families
    data BuiltInSynFamily = BuiltInSynFamily
      { sfMatchFam      :: [Type] -> Maybe (CoAxiomRule, [Type], Type)
      , sfInteractTop   :: [Type] -> Type -> [Eqn]
      , sfInteractInert :: [Type] -> Type ->
                           [Type] -> Type -> [Eqn]
      }

    -- Provides default implementations that do nothing.
    trivialBuiltInFamily :: BuiltInSynFamily
    trivialBuiltInFamily = BuiltInSynFamily
      { sfMatchFam      = \_ -> Nothing
      , sfInteractTop   = \_ _ -> []
      , sfInteractInert = \_ _ _ _ -> []
      }