Vedanta

Type-level reasoning

A function application - corresponds to /Modus Ponens/
A specialization, an instance of it.

> ($) :: (a -> b) -> a -> b
> ($) f x = f x

Reversed order of agreements, still the same /Modus Ponens/

> (|>) ::  a -> (a -> b) -> b
> (|>) x f = f x

"Multi-argument" is just /currying/ -- ~* ->*->*-> *~

Same, but with an explicit /abstraction barrier/ (and /lifting/)

> (>>=) :: m a -> (a -> m b) -> m b
> (>>=) ma f = undefined  -- has to be defined for each particular type (an instance)

Notice that this is essentially a type-signature of ~fmap~,

> fmap :: f a ->(a->b) -> f b

with a reversed order of arguments and a specialized function.
which corresponds to a /Kleisli arrow/.

Notice also that ~($)~ and ~fmap~ differs only in having an explicit /container/.
This is not an arbitrary coincidence.

All three are essentially the same, despite funny syntax.
A basic step of logical deduction. Behind an abstraction barrier.

Composition

- of functions

> (.) :: (b -> c) -> (a -> b) -> a -> c
> (.) f g = \x -> f (g x)

- of "actions" (actual functions have to use ~return :: a -> m a~)

> (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c
> (<=<) f g = undefined -- has to be defined for each particular type

or ~(>>=)~ for chaining ("flat" nesting or pipelining ) , exactly the same as ~(|>)~

Notice that the composition is /nesting/. This is the /only/ way to implement it.

The end of knowledge.