    -- | A strict version of the semi-direct product. If a monoid m acts
    --   on s then this version of the semi-direct product is strict in
    --   the m-portion of the semi-direct product.

    {-# LANGUAGE CPP                   #-}
    {-# LANGUAGE FlexibleInstances     #-}
    {-# LANGUAGE MultiParamTypeClasses #-}
    {-# LANGUAGE TupleSections         #-}

    module Data.Monoid.SemiDirectProduct.Strict
           ( Semi, unSemi, tag, inject, untag, embed, quotient
           ) where

    #if !MIN_VERSION_base(4,8,0)
    import           Data.Monoid        (Monoid(..))
    #endif
    import           Data.Semigroup     (Semigroup(..))

    import           Data.Monoid.Action

    -- | The semi-direct product of monoids @s@ and @m@, which is a monoid
    --   when @m@ acts on @s@. Structurally, the semi-direct product is
    --   just a pair @(s,m)@.  However, the monoid instance is different.
    --   In particular, we have
    --
    -- > (s1,m1) <> (s2,m2) = (s1 <> (m1 `act` s2), m1 <> m2)
    --
    --   We call the monoid @m@ the quotient monoid and the monoid @s@ the
    --   sub-monoid of the semi-direct product. The semi-direct product
    --   @Semi s m@ is an extension of the monoid @s@ with @m@ being the
    --   quotient.
    data Semi s m = Semi s !m

    unSemi :: Semi s m -> (s,m)
    unSemi :: forall s m. Semi s m -> (s, m)
    unSemi (Semi s
    s m
    m) = (s
    s,m
    m)

    instance (Semigroup m, Semigroup s, Action m s) => Semigroup (Semi s m) where
      Semi s
    xs m
    xm <> :: Semi s m -> Semi s m -> Semi s m
    <> Semi s
    ys m
    ym          = s -> m -> Semi s m
    forall s m. s -> m -> Semi s m
    Semi (s
    xs s -> s -> s
    forall a. Semigroup a => a -> a -> a
    <> (m
    xm m -> s -> s
    forall m s. Action m s => m -> s -> s
    `act` s
    ys)) (m
    xm m -> m -> m
    forall a. Semigroup a => a -> a -> a
    <> m
    ym)
      {-# INLINE (<>) #-}

    #if MIN_VERSION_base(4,8,0)
      sconcat :: NonEmpty (Semi s m) -> Semi s m
    sconcat                           = (Semi s m -> Semi s m -> Semi s m)
    -> NonEmpty (Semi s m) -> Semi s m
    forall a. (a -> a -> a) -> NonEmpty a -> a
    forall (t :: * -> *) a. Foldable t => (a -> a -> a) -> t a -> a
    foldr1 Semi s m -> Semi s m -> Semi s m
    forall a. Semigroup a => a -> a -> a
    (<>)
      {-# INLINE sconcat #-}
    #endif

    instance (Monoid m, Monoid s, Action m s) => Monoid (Semi s m) where
      mempty :: Semi s m
    mempty                            = s -> m -> Semi s m
    forall s m. s -> m -> Semi s m
    Semi s
    forall a. Monoid a => a
    mempty m
    forall a. Monoid a => a
    mempty
      {-# INLINE mempty #-}
    #if !MIN_VERSION_base(4,11,0)
      mappend (Semi xs xm) (Semi ys ym) = Semi (xs `mappend` (xm `act` ys)) (xm `mappend` ym)
      {-# INLINE mappend #-}
    #endif
      mconcat :: [Semi s m] -> Semi s m
    mconcat                           = (Semi s m -> Semi s m -> Semi s m)
    -> Semi s m -> [Semi s m] -> Semi s m
    forall a b. (a -> b -> b) -> b -> [a] -> b
    forall (t :: * -> *) a b.
    Foldable t =>
    (a -> b -> b) -> b -> t a -> b
    foldr Semi s m -> Semi s m -> Semi s m
    forall a. Monoid a => a -> a -> a
    mappend Semi s m
    forall a. Monoid a => a
    mempty
      {-# INLINE mconcat #-}

    -- | Tag an @s@ value with an @m@ value to create an element of the
    --   semi-direct product.
    tag :: s -> m -> Semi s m
    tag :: forall s m. s -> m -> Semi s m
    tag = s -> m -> Semi s m
    forall s m. s -> m -> Semi s m
    Semi

    -- | The injection map, /i.e./ give an @s@ value a trivial tag.
    inject :: Monoid m => s -> Semi s m
    inject :: forall m s. Monoid m => s -> Semi s m
    inject = (s -> m -> Semi s m) -> m -> s -> Semi s m
    forall a b c. (a -> b -> c) -> b -> a -> c
    flip s -> m -> Semi s m
    forall s m. s -> m -> Semi s m
    Semi m
    forall a. Monoid a => a
    mempty

    -- | Forget the monoidal tag.  Of course, @untag . inject = id@, and
    --   @untag (tag s m) = s@.
    untag :: Semi s m -> s
    untag :: forall s m. Semi s m -> s
    untag (Semi s
    s m
    _) = s
    s

    -- | Embed a "tag" value as a value of type @Semi s m@.  Note that
    --
    --   @inject s <> embed m = tag s m@
    --
    --   and
    --
    --   @embed m <> inject s@ = tag (act m s) m@
    --
    --   The semi-direct product gives a split extension of @s@ by
    --   @m@. This allows us to embed @m@ into the semi-direct
    --   product. This is the embedding map. The quotient and embed maps
    --   should satisfy the equation @quotient . embed = id@.
    embed :: Monoid s => m -> Semi s m
    embed :: forall s m. Monoid s => m -> Semi s m
    embed = s -> m -> Semi s m
    forall s m. s -> m -> Semi s m
    Semi s
    forall a. Monoid a => a
    mempty

    -- | The quotient map, /i.e./ retrieve the monoidal tag value.
    quotient :: Semi s m -> m
    quotient :: forall s m. Semi s m -> m
    quotient (Semi s
    _ m
    m) = m
    m