# Running example: Addition of Natural Numbers

Throughout this tutorial, we will be working with the following function, defined in the Idris prelude, which defines addition on natural numbers:

```
plus : Nat -> Nat -> Nat
plus Z m = m
plus (S k) m = S (plus k m)
```

It is defined by the above equations, meaning that we have for free the
properties that adding `m`

to zero always results in `m`

, and that
adding `m`

to any non-zero number `S k`

always results in
`S (plus k m)`

. We can see this by evaluation at the Idris REPL (i.e.
the prompt, the read-eval-print loop):

```
Main> \m => plus Z m
\m => m
Idris> \k,m => plus (S k) m
\k => \m => S (plus k m)
```

Note that unlike many other language REPLs, the Idris REPL performs
evaluation on *open* terms, meaning that it can reduce terms which
appear inside lambda bindings, like those above. Therefore, we can
introduce unknowns `k`

and `m`

as lambda bindings and see how
`plus`

reduces.

The `plus`

function has a number of other useful properties, for
example:

It is

*commutative*, that is for all`Nat`

inputs`n`

and`m`

, we know that`plus n m = plus m n`

.It is

*associative*, that is for all`Nat`

inputs`n`

,`m`

and`p`

, we know that`plus n (plus m p) = plus (plus m n) p`

.

We can use these properties in an Idris program, but in order to do so
we must *prove* them.

## Equality Proofs

Idris defines a propositional equality type as follows:

```
data Equal : a -> b -> Type where
Refl : Equal x x
```

As syntactic sugar, `Equal x y`

can be written as `x = y`

.

It is *propositional* equality, where the type states that any two
values in different types `a`

and `b`

may be proposed to be equal.
There is only one way to *prove* equality, however, which is by
reflexivity (`Refl`

).

We have a *type* for propositional equality here, and correspondingly a
*program* inhabiting an instance of this type can be seen as a proof of
the corresponding proposition [1]. So, trivially, we can prove that
`4`

equals `4`

:

```
four_eq : 4 = 4
four_eq = Refl
```

However, trying to prove that `4 = 5`

results in failure:

```
four_eq_five : 4 = 5
four_eq_five = Refl
```

The type `4 = 5`

is a perfectly valid type, but is uninhabited, so
when trying to type check this definition, Idris gives the following
error:

```
When unifying 4 = 4 and (fromInteger 4) = (fromInteger 5)
Mismatch between:
4
and
5
```

### Type checking equality proofs

An important step in type checking Idris programs is *unification*,
which attempts to resolve implicit arguments such as the implicit
argument `x`

in `Refl`

. As far as our understanding of type checking
proofs is concerned, it suffices to know that unifying two terms
involves reducing both to normal form then trying to find an assignment
to implicit arguments which will make those normal forms equal.

When type checking `Refl`

, Idris requires that the type is of the form
`x = x`

, as we see from the type of `Refl`

. In the case of
`four_eq_five`

, Idris will try to unify the expected type `4 = 5`

with the type of `Refl`

, `x = x`

, notice that a solution requires
that `x`

be both `4`

and `5`

, and therefore fail.

Since type checking involves reduction to normal form, we can write the following equalities directly:

```
twoplustwo_eq_four : 2 + 2 = 4
twoplustwo_eq_four = Refl
plus_reduces_Z : (m : Nat) -> plus Z m = m
plus_reduces_Z m = Refl
plus_reduces_Sk : (k, m : Nat) -> plus (S k) m = S (plus k m)
plus_reduces_Sk k m = Refl
```

## Heterogeneous Equality

Equality in Idris is *heterogeneous*, meaning that we can even propose
equalities between values in different types:

```
idris_not_php : Z = "Z"
```

The type `Z = "Z"`

is uninhabited, and one might wonder why it is useful to
be able to propose equalities between values in different types. However, with
dependent types, such equalities can arise naturally. For example, if two
vectors are equal, their lengths must be equal:

```
vect_eq_length : (xs : Vect n a) -> (ys : Vect m a) ->
(xs = ys) -> n = m
```

In the above declaration, `xs`

and `ys`

have different types because
their lengths are different, but we would still like to draw a
conclusion about the lengths if they happen to be equal. We can define
`vect_eq_length`

as follows:

```
vect_eq_length xs xs Refl = Refl
```

By matching on `Refl`

for the third argument, we know that the only
valid value for `ys`

is `xs`

, because they must be equal, and
therefore their types must be equal, so the lengths must be equal.

Alternatively, we can put an underscore for the second `xs`

, since
there is only one value which will type check:

```
vect_eq_length xs _ Refl = Refl
```

## Properties of `plus`

Using the `(=)`

type, we can now state the properties of `plus`

given above as Idris type declarations:

```
plus_commutes : (n, m : Nat) -> plus n m = plus m n
plus_assoc : (n, m, p : Nat) -> plus n (plus m p) = plus (plus n m) p
```

Both of these properties (and many others) are proved for natural number
addition in the Idris standard library, using `(+)`

from the `Num`

interface rather than using `plus`

directly. They have the names
`plusCommutative`

and `plusAssociative`

respectively.

In the remainder of this tutorial, we will explore several different
ways of proving `plus_commutes`

(or, to put it another way, writing
the function.) We will also discuss how to use such equality proofs, and
see where the need for them arises in practice.