Views and the “with” rule

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:

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 (succ n)
  foo _ m | 2 with (succ m)
    foo _ _ | 2 | 3 = True
    foo _ _ | 2 | _ = False
  foo _ _ | _ = 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 : Parity (n + n)
   Odd  : 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.

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.

We will return to this function in the next section Theorems in Practice to complete the definition of parity.

With and proofs

To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match. To do this, you can postfix the with clause with proof p and the proof generated by the pattern match will be in scope and named p. For example:

data Foo = FInt Int | FBool Bool

optional : Foo -> Maybe Int
optional (FInt x) = Just x
optional (FBool b) = Nothing

isFInt : (foo:Foo) -> Maybe (x : Int ** (optional foo = Just x))
isFInt foo with (optional foo) proof p
  isFInt foo | Nothing = Nothing           -- here, p : Nothing = optional foo
  isFInt foo | (Just x) = Just (x ** Refl) -- here, p : Just x = optional foo

[1]	Conor McBride and James McKinna. 2004. The view from the left. J. Funct. Program. 14, 1 (January 2004), 69-111. DOI=10.1017/S0956796803004829 http://dx.doi.org/10.1017/S0956796803004829ñ