`mid (/εmaidεnt@tI/ aka unit, return, pure): P -> `__P__

`map (/maep/): (P -> Q) -> `__P__ -> __Q__

-`map2 (/maeptu/): (P -> Q -> R) -> `__P__ -> __Q__ -> __R__

+`map2 (/m&ash;ptu/): (P -> Q -> R) -> `__P__ -> __Q__ -> __R__

`mapply (/εm@plai/): `__P -> Q__ -> __P__ -> __Q__

@@ -108,8 +109,8 @@ if there is a `map` function defined for that box type with the type given above
if there are in addition `map2`, `mid`, and `mapply`. (With
`map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.)
-* ***Monad*** ("composable") A MapNable box type is a *Monad* if there
- is in addition an `mcompose` and a `join` such that `mid` is be
+* ***Monad*** ("composables") A MapNable box type is a *Monad* if there
+ is in addition an `mcompose` and a `join` such that `mid` is
a left and right identity for `mcompose`, and `mcompose` is
associative. That is, the following "laws" must hold:
@@ -124,26 +125,26 @@ Identity box type is a completly invisible box. With the following
definitions
mid â¡ \p.p
- mcompose â¡ \f\g\x.f(gx)
+ mcompose â¡ \fgx.f(gx)
Id is a monad. Here is a demonstration that the laws hold:
- mcompose mid k == (\f\g\x.f(gx)) (\p.p) k
+ mcompose mid k == (\fgx.f(gx)) (\p.p) k
~~> \x.(\p.p)(kx)
~~> \x.kx
~~> k
- mcompose k mid == (\f\g\x.f(gx)) k (\p.p)
+ mcompose k mid == (\fgx.f(gx)) k (\p.p)
~~> \x.k((\p.p)x)
~~> \x.kx
~~> k
- mcompose (mcompose j k) l == mcompose ((\f\g\x.f(gx)) j k) l
+ mcompose (mcompose j k) l == mcompose ((\fgx.f(gx)) j k) l
~~> mcompose (\x.j(kx)) l
- == (\f\g\x.f(gx)) (\x.j(kx)) l
+ == (\fgx.f(gx)) (\x.j(kx)) l
~~> \x.(\x.j(kx))(lx)
~~> \x.j(k(lx))
- mcompose j (mcompose k l) == mcompose j ((\f\g\x.f(gx)) k l)
+ mcompose j (mcompose k l) == mcompose j ((\fgx.f(gx)) k l)
~~> mcompose j (\x.k(lx))
- == (\f\g\x.f(gx)) j (\x.k(lx))
+ == (\fgx.f(gx)) j (\x.k(lx))
~~> \x.j((\x.k(lx)) x)
~~> \x.j(k(lx))
@@ -157,10 +158,15 @@ consider the box type `Î± List`, with the following operations:
mcompose-crossy: (Î² -> [Î³]) -> (Î± -> [Î²]) -> (Î± -> [Î³])
mcompose-crossy f g a = [c | b <- g a, c <- f b]
+ mcompose-crossy f g a = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
+ mcompose-crossy f g a = concat (map f (g a))
+These three definitions are all equivalent.
In words, `mcompose f g a` feeds the a (which has type Î±) to g, which
returns a list of Î²s; each Î² in that list is fed to f, which returns a
-list of Î³s. The final result is the concatenation of those lists of Î³s.
+list of Î³s.
+
+The final result is the concatenation of those lists of Î³s.
For example,
let f b = [b, b+1] in
@@ -169,12 +175,11 @@ For example,
It is easy to see that these definitions obey the monad laws (see exercises).
-There can be multiple monads for any given box type. For isntance,
+There can be multiple monads for any given box type. For instance,
using the same box type and the same mid, we can define
- mcompose-zippy f g a = match (f,g) with
- ([],_) -> []
- (_,[]) -> []
- (f:ftail, g:gtail) -> f(ga) && mcompoze-zippy ftail gtail a
+ mcompose-zippy f g a = foldr (\b -> \gs -> f b ++ gs) (g a) []
+so that
+ mcompose-zippy f g 7 = [49, 14]
--
2.11.0