-- This file:
--   http://angg.twu.net/HASKELL/Applicative1.hs.html
--   http://angg.twu.net/HASKELL/Applicative1.hs
--           (find-angg "HASKELL/Applicative1.hs")
-- Author: Eduardo Ochs <eduardoochs@gmail.com>
--
-- (defun e () (interactive) (find-angg "HASKELL/Applicative1.hs"))
-- (find-es "haskell" "applicative")
-- (find-es "haskell" "Prelude")
-- (find-es "haskell" "InstanceSigns")
-- (find-books "__comp/__comp.el" "haskell-hutton")
-- (find-books "__comp/__comp.el" "haskell-hutton" "Applicatives")
-- https://www.staff.city.ac.uk/~ross/papers/Applicative.pdf
-- https://www.staff.city.ac.uk/~ross/papers/Applicative.html



-- (find-huttonbookpage 182 "12.1     Functors")
-- (find-huttonbooktext 182 "12.1     Functors")
-- (find-ghcbasefile "Data/Functor.hs")
-- (find-ghcbasefile "Data/Functor.hs" "An infix synonym for 'fmap'")
--
-- class Functor f where
--   fmap :: (a -> b) -> f a -> f b


-- (find-huttonbookpage 187 "12.2     Applicatives")
-- (find-huttonbooktext 187 "12.2     Applicatives")
-- (find-ghcbasefile "Control/Applicative.hs")
--
-- class Functor f => Applicative f where
--    pure :: a -> f a
--    (<*>) :: f (a -> b) -> f a -> f b
-- 
-- fmap0 :: a -> f a
-- fmap0 = pure
-- 
-- fmap1 :: (a -> b) -> f a -> f b
-- fmap1 g x = pure g <*> x
-- 
-- fmap1  g    x   =  pure  g   <*>  x
--       ==== ===          ====     ===
--       a->b f a          a->b     f a
-- ==========         =========
--   f a->f b           f(a->b)
-- ===============    =================
--        f b                   f b
-- 
-- fmap2 :: (a -> b -> c) -> f a -> f b -> f c
-- fmap2 g x y = pure g <*> x <*> y
-- 
-- fmap2    g     x   y   = pure    g    <*>  x  <*>  y
--       ======= === ===         =======     ===     ===
--       a->b->c f a f b         a->b->c     f a     f b
-- =============            ============
-- f a->f b->f c              f(a->b->c)
-- =================        ====================
--      f b->f c                 f(b->c)
-- =====================    ============================
--           f c                    f c
-- 
-- fmap3 :: (a -> b -> c -> d) -> f a -> f b -> f c -> f d
-- fmap3 g x y z = pure g <*> x <*> y <*> z
--
-- fmap3        g      x   y   z   =  pure      g     <*>  x <*>  y <*>  z 
--         ========== === === ===          ==========     ===    ===    ===
--         a->b->c->d f a f b f c          a->b->c->d     f a    f b    f c
-- ==================                 ===============                      
-- f a->f b->f c->f d                   f(a->b->c->d)                      
-- ======================             =======================              
--    f b ->f c-> f d                      f(b->c->d)                      
-- ==========================         ==============================       
--           f c->f d                         f(c->d)                      
-- ==============================     =====================================
--                f d                            f d                       



-- (find-huttonbookpage 193 "12.3      Monads")
-- (find-huttonbooktext 193 "12.3      Monads")
--
-- class Applicative m => Monad m where
--    return :: a -> m a
--    (>>=) :: m a -> (a -> m b) -> m b
--    return = pure



-- liftM f    mx  ==  mx  >>= \x -> return ( f   x)
--       ==== ===     ===      =            ==== =
--       a->b m a     m a      a            a->b a
--                                          ======
--                                             b
--                                  ===============
--                                       m b
--                            =====================
--                                    a->m b
--                    =============================
--                                       m b


f <$> mx ={synonym}=
  fmap f mx ={law}=
    liftM f mx ={definition}=
                mx >>= \x -> return (f x)

                                             to
      simplify the `m1 >>= \x1 -> return (f x0 x1)' to `f x0 <$>
      m1')


f <$> mx
= fmap f mx
= liftM f mx
= mx >>= \x -> return (f x)

  = m1 >>= \x1 -> return (f x0 x1)




fmap :: (a -> b) -> (f a -> f b)
(<$>) = fmap
(<$> mx) = \ f   -> fmap  f   mx
            ====         ==== ===
            a->b         a->b ma
                    =============
                        m b

<ski> mf `ap` mx = mf >>= \f -> f <$> mx = mf >>= (<$> mx)

mf `ap` mx
= ap mf mx
= (<*>) mf mx
= mf >>= \f -> f <$> mx
= mf >>= (<$> mx)


(find-ghcbasefile "GHC/Base.hs" "Promote a function to a monad")
(find-ghcbasefile "GHC/Base.hs" "Promote a function to a monad" "liftM2")
(find-ghcbasefile "GHC/Base.hs" "(<*>) = liftA2 id")



<lambdabot> ap = liftM2 id
<hpc> bah
<ski> well, i pronounce it `ap' .. for the other question, see ^
<hpc> anyhoo, ap = (<*>), so see how that's defined
<ski> @src liftM2
<lambdabot> liftM2 f m1 m2 = do
<lambdabot> x1 <- m1
<lambdabot> x2 <- m2
<lambdabot> return (f x1 x2)
<ski> liftM2 f m0 m1 = m0 >>= \x0 -> m1 >>= \x1 -> return (f x0 x1)
<ski> = m0 >>= \x0 -> (return . f x0 <$> m1)
<ski> so
<edrx> so that's a definition for <*>?
*** chexum JOIN
<edrx> I'll need some minutes to type each subexpression of that
<geekosaur> use `id` for `f`
<ski> er, sorry, no `return' in the last line
<ski> = m0 >>= \x0 -> (f x0 <$> m1)
<ski> mf `ap` mx = mf >>= \f -> f <$> mx
<ski> = mf >>= (<$> mx)
<ski> (i used here that `f <$> mx ={synonym}= fmap f mx ={law}=
      liftM f mx ={definition}= mx >>= \x -> return (f x)' to
      simplify the `m1 >>= \x1 -> return (f x0 x1)' to `f x0 <$>
      m1')


  

-- (<*>)   mf   mx  =   mf    >>= \ f   -> fmap  f    mx
--      ======= ===   =======      ====          ==== ===
--      m(a->b) m a   m(a->b)      a->b          a->b m a
-- ================                        ==============
--        m b                                     m b
--                                =======================
--                                        (a->b)->m b
--                    ==================================
--                                                m b

    
     
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