# Views and the “`with`

” rule

Warning

NOT UPDATED FOR IDRIS 2 YET

## Dependent pattern matching

Since types can depend on values, the form of some arguments can be
determined by the value of others. For example, if we were to write
down the implicit length arguments to `(++)`

, we’d see that the form
of the length argument was determined by whether the vector was empty
or not:

```
(++) : Vect n a -> Vect m a -> Vect (n + m) a
(++) {n=Z} [] ys = ys
(++) {n=S k} (x :: xs) ys = x :: xs ++ ys
```

If `n`

was a successor in the `[]`

case, or zero in the `::`

case, the definition would not be well typed.

## The `with`

rule — matching intermediate values

Very often, we need to match on the result of an intermediate
computation. Idris provides a construct for this, the `with`

rule, inspired by views in `Epigram`

[1], which takes account of
the fact that matching on a value in a dependently typed language can
affect what we know about the forms of other values. In its simplest
form, the `with`

rule adds another argument to the function being
defined.

We have already seen a vector filter function. This time, we define it
using `with`

as follows:

```
filter : (a -> Bool) -> Vect n a -> (p ** Vect p a)
filter p [] = ( _ ** [] )
filter p (x :: xs) with (filter p xs)
filter p (x :: xs) | ( _ ** xs' ) = if (p x) then ( _ ** x :: xs' ) else ( _ ** xs' )
```

Here, the `with`

clause allows us to deconstruct the result of
`filter p xs`

. The view refined argument pattern ```
filter p (x ::
xs)
```

goes beneath the `with`

clause, followed by a vertical bar
`|`

, followed by the deconstructed intermediate result ```
( _ ** xs'
)
```

. If the view refined argument pattern is unchanged from the
original function argument pattern, then the left side of `|`

is
extraneous and may be omitted with an underscore `_`

:

```
filter p (x :: xs) with (filter p xs)
_ | ( _ ** xs' ) = if (p x) then ( _ ** x :: xs' ) else ( _ ** xs' )
```

`with`

clauses can also be nested:

```
foo : Int -> Int -> Bool
foo n m with (n + 1)
foo _ m | 2 with (m + 1)
foo _ _ | 2 | 3 = True
foo _ _ | 2 | _ = False
foo _ _ | _ = False
```

and left hand sides that are the same as their parent’s can be skipped by
using `_`

to focus on the patterns for the most local `with`

. Meaning
that the above `foo`

can be rewritten as follows:

```
foo : Int -> Int -> Bool
foo n m with (n + 1)
_ | 2 with (m + 1)
_ | 3 = True
_ | _ = False
_ | _ = False
```

If the intermediate computation itself has a dependent type, then the
result can affect the forms of other arguments — we can learn the form
of one value by testing another. In these cases, view refined argument
patterns must be explicit. For example, a `Nat`

is either even or
odd. If it is even it will be the sum of two equal `Nat`

.
Otherwise, it is the sum of two equal `Nat`

plus one:

```
data Parity : Nat -> Type where
Even : {n : _} -> Parity (n + n)
Odd : {n : _} -> Parity (S (n + n))
```

We say `Parity`

is a *view* of `Nat`

. It has a *covering function*
which tests whether it is even or odd and constructs the predicate
accordingly. Note that we’re going to need access to `n`

at run time, so
although it’s an implicit argument, it has unrestricted multiplicity.

```
parity : (n:Nat) -> Parity n
```

We’ll come back to the definition of `parity`

shortly. We can use it
to write a function which converts a natural number to a list of
binary digits (least significant first) as follows, using the `with`

rule:

```
natToBin : Nat -> List Bool
natToBin Z = Nil
natToBin k with (parity k)
natToBin (j + j) | Even = False :: natToBin j
natToBin (S (j + j)) | Odd = True :: natToBin j
```

The value of `parity k`

affects the form of `k`

, because the
result of `parity k`

depends on `k`

. So, as well as the patterns
for the result of the intermediate computation (`Even`

and `Odd`

)
right of the `|`

, we also write how the results affect the other
patterns left of the `|`

. That is:

When

`parity k`

evaluates to`Even`

, we can refine the original argument`k`

to a refined pattern`(j + j)`

according to`Parity (n + n)`

from the`Even`

constructor definition. So`(j + j)`

replaces`k`

on the left side of`|`

, and the`Even`

constructor appears on the right side. The natural number`j`

in the refined pattern can be used on the right side of the`=`

sign.Otherwise, when

`parity k`

evaluates to`Odd`

, the original argument`k`

is refined to`S (j + j)`

according to`Parity (S (n + n))`

from the`Odd`

constructor definition, and`Odd`

now appears on the right side of`|`

, again with the natural number`j`

used on the right side of the`=`

sign.

Note that there is a function in the patterns (`+`

) and repeated
occurrences of `j`

- this is allowed because another argument has
determined the form of these patterns.

## Defining `parity`

The definition of `parity`

is a little tricky, and requires some knowledge of
theorem proving (see Section Theorem Proving), but for completeness, here
it is:

```
parity : (n : Nat) -> Parity n
parity Z = Even {n = Z}
parity (S Z) = Odd {n = Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even
= rewrite plusSuccRightSucc j j in Even {n = S j}
parity (S (S (S (j + j)))) | Odd
= rewrite plusSuccRightSucc j j in Odd {n = S j}
```

For full details on `rewrite`

in particular, please refer to the theorem
proving tutorial, in Section Theorem Proving.