In the last post we saw that we can compose an effectful program `f: (a: A) => F<B>`

with a pure program `g: (b: B) => C`

by lifting `g`

to a function `lift(g): (fb: F<B>) => F<C>`

provided that `F`

admits a functor instance

Program f | Program g | Composition |
---|---|---|

pure | pure | `g ∘ f` |

effectful | pure (unary) | `lift(g) ∘ f` |

However `g`

must be unary, that is it must accept only one argument as input. What if `g`

accepts two arguments? Can we still lift `g`

by using only the functor instance? Well, let's try!

# Currying

First of all we must model a function that accepts two arguments, let's say of type `B`

and `C`

(we can use a tuple) and returns a value of type `D`

```
g: (args: [B, C]) => D
```

We can rewrite `g`

using a technique called **currying**.

Currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions,

each with a single argument. For example, a function that takes two arguments, one from`B`

and one from`C`

, and produces outputs in`D`

, by currying is translated into a function that takes a single argument from`C`

and produces as outputs functions from`B`

to`C`

.

(source: currying on wikipedia.org)

So we can rewrite `g`

to

```
g: (b: B) => (c: C) => D
```

What we want is a lifting operation, let't call it `liftA2`

in order to distinguish it from our old `lift`

, that outputs a function with the following signature

```
liftA2(g): (fb: F<B>) => (fc: F<C>) => F<D>
```

How can we get there? Since `g`

is now unary, we can use the functor instance and our old `lift`

```
lift(g): (fb: F<B>) => F<(c: C) => D>
```

But now we are stuck: there's no legal operation on the functor instance which is able to **unpack** the value `F<(c: C) => D>`

to a function `(fc: F<C>) => F<D>`

.

# Apply

So let's introduce a new abstraction `Apply`

that owns such a unpacking operation (named `ap`

)

```
interface Apply<F> extends Functor<F> {
ap: <C, D>(fcd: HKT<F, (c: C) => D>, fc: HKT<F, C>) => HKT<F, D>
}
```

The `ap`

function is basically `unpack`

with the arguments rearranged

```
unpack: <C, D>(fcd: HKT<F, (c: C) => D>) => ((fc: HKT<F, C>) => HKT<F, D>)
ap: <C, D>(fcd: HKT<F, (c: C) => D>, fc: HKT<F, C>) => HKT<F, D>
```

so `ap`

can be derived from `unpack`

(and viceversa).

**Note**: the `HKT`

type is the `fp-ts`

way to represent a generic type constructor (a technique proposed in the Lightweight higher-kinded polymorphism paper) so when you see `HKT<F, X>`

you can think to the type constructor `F`

applied to the type `X`

(i.e. `F<X>`

).

# Applicative

Moreover it would be handy if there exists an operation which is able to **lift a value** of type `A`

to a value of type `F<A>`

. This way we could call the `liftA2(g)`

function either by providing arguments of type `F<B>`

and `F<C>`

or by lifting values of type `B`

and `C`

.

So let's introduce the `Applicative`

abstraction which builds upon `Apply`

and owns such operation (named `of`

)

```
interface Applicative<F> extends Apply<F> {
of: <A>(a: A) => HKT<F, A>
}
```

Let's see the `Applicative`

instances for some common data types

**Example** (`F = Array`

)

```
import { flatten } from 'fp-ts/lib/Array'
const applicativeArray = {
map: <A, B>(fa: Array<A>, f: (a: A) => B): Array<B> => fa.map(f),
of: <A>(a: A): Array<A> => [a],
ap: <A, B>(fab: Array<(a: A) => B>, fa: Array<A>): Array<B> =>
flatten(fab.map(f => fa.map(f)))
}
```

**Example** (`F = Option`

)

```
import { Option, some, none, isNone } from 'fp-ts/lib/Option'
const applicativeOption = {
map: <A, B>(fa: Option<A>, f: (a: A) => B): Option<B> =>
isNone(fa) ? none : some(f(fa.value)),
of: <A>(a: A): Option<A> => some(a),
ap: <A, B>(fab: Option<(a: A) => B>, fa: Option<A>): Option<B> =>
isNone(fab) ? none : applicativeOption.map(fa, fab.value)
}
```

**Example** (`F = Task`

)

```
import { Task } from 'fp-ts/lib/Task'
const applicativeTask = {
map: <A, B>(fa: Task<A>, f: (a: A) => B): Task<B> => () => fa().then(f),
of: <A>(a: A): Task<A> => () => Promise.resolve(a),
ap: <A, B>(fab: Task<(a: A) => B>, fa: Task<A>): Task<B> => () =>
Promise.all([fab(), fa()]).then(([f, a]) => f(a))
}
```

# Lifting

So given an instance of `Apply`

for `F`

can we now write `liftA2`

?

```
import { HKT } from 'fp-ts/lib/HKT'
import { Apply } from 'fp-ts/lib/Apply'
type Curried2<B, C, D> = (b: B) => (c: C) => D
function liftA2<F>(
F: Apply<F>
): <B, C, D>(g: Curried2<B, C, D>) => Curried2<HKT<F, B>, HKT<F, C>, HKT<F, D>> {
return g => fb => fc => F.ap(F.map(fb, g), fc)
}
```

Nice! But what about functions with **three** arguments? Do we need *yet another abstraction*?

The good news is that the answer is no, `Apply`

is enough

```
type Curried3<B, C, D, E> = (b: B) => (c: C) => (d: D) => E
function liftA3<F>(
F: Apply<F>
): <B, C, D, E>(
g: Curried3<B, C, D, E>
) => Curried3<HKT<F, B>, HKT<F, C>, HKT<F, D>, HKT<F, E>> {
return g => fb => fc => fd => F.ap(F.ap(F.map(fb, g), fc), fd)
}
```

Actually given an instance of `Apply`

we can write a `liftAn`

function, **for each** `n`

.

**Note**. `liftA1`

is just `lift`

, the `Functor`

operation.

We can now update our "composition table"

Program f | Program g | Composition |
---|---|---|

pure | pure | `g ∘ f` |

effectful | pure, `n` -ary |
`liftAn(g) ∘ f` |

`liftA1 = lift`

# Is the general problem solved?

Not yet. There's still an important case which is missing: what if **both** programs are effectful?

Again we need something more: in the next post I'll talk about one of the most important abstractions of functional programming: **monads**.

TLDR: functional programming is all about composition

Posted on by:

## Discussion

Last year I read a 700-pages book about Haskell - and I never got the feeling of

actuallyunderstanding what is going on (like even the basic concepts).And reading these articles for real kicked me in - I don't know if I just needed another try to align the knowledge in my head or you are just

that goodat explaining things - either way, my eternal gratitude.