# Pi Types

We will try to formulate and prove the statement 
> The sum of two even naturals is even.
To do so we must define `+` on the naturals.
Addition takes in two naturals and spits out a
natural, so it should have type `ℕ → ℕ → ℕ`.
```agda
_+_ : ℕ → ℕ → ℕ
n + m = ?
```
Agda supports the notation `_+_` (without spaces)
which means from now on you can write `0 + 1` 
and so on (with spaces).
Try coming up with a sensible definition yourself,
it may not look exactly like ours.
<details>
  <summary>Hint</summary>
  `n + 0` should be `n` and `n + (m + 1)` should be `(n + m) + 1`
</details>

Now we can make the statement:
```agda
SumOfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst + y .fst)
SumOfEven x y = ?
```
> Tip: `x .fst` is another notation for `fst x`.
> This works for all sigma types.
There are three ways to interpret this:
- For all even naturals `x` and for all even naturals `y`, 
  their sum is even.
- `isEven (x .fst + y .fst)` is a construction depending on two recipes
  `x` and `y`.
  Given two recipes `x` and `y` of `Σ ℕ isEven`, 
  we break them down into their first components,
  apply the conversion `_+_`,
  and form a recipe for `isEven` of the result.
- `isEven (_ .fst + _ .fst)` is a bundle over the categorical product
  `Σ ℕ isEven × Σ ℕ isEven` and `SumOfEven` is a _section_ of the bundle.
  
More generally given `A : Type` and `B : A → Type` we can form the _pi type_
`(x : A) → B x : Type` (in other languages `Π (x : ℕ), isEven n`). 
The notation suggests that these behave like functions,
and indeed in the special case where the fiber is constant 
with respect to the base a section is just a term of type `A → B`,
i.e. a function. Hence pi types are also known as 
_dependent function types_.

We are now in a position to prove the statement. Have fun!

_Important_: Once you have proven the statement, 
check out our two ways of defining addition `_+_` and `_+'_`
(in the solutions).
Use `C-c C-n` to check that they compute the same values
on different examples.
Uncomment the code for `Sum'OfEven` in the solutions,
it is just `SumOfEven` but with `+`s changed for `+'`s.
Load the file. Does the proof still work?

In our proof of `SumOfEven` we explicitely used the definition of `_+_`,
which means that if we wanted to use our proof on someone else's 
definition of addition, things might break.
> But `_+_` and `_+'_` compute the same values. 
> Are `_+_` and `_+'_` 'the same'? What is 'the same'?