## Desugaring of array comprehensions

Wadler's desugaring for list comprehensions is not suitable for arrays, as we need to use collective operations to get good parallel code. The build/foldr desugaring, although using collective operations, isn't a good match for how the array operations are implemented. In fact, the *naive* desugaring from the H98 report is a much better fit:

```
(1) [: e | :] = [:e:]
(2) [: e | b, qs :] = if b then [: e | qs :] else [::]
(3) [: e | p <- a, qs :] = let ok p = [: e | qs :]
ok _ = [::]
in concatMapP ok a
(4) [: e | let ds, qs :] = let ds in [: e | qs :]
(5) [: e | qs | qss :] =
[: e | (XS, XSS) <- zip [: XS | qs :] [: XSS | qss :] :]
where XS & XSS are the bound variables in qs & qss
```

In particular, `concatMapP f a`

essentially implies to apply the lifted version of `f`

directly to `a`

and then the concat strips of one level of segment descriptors; i.e., both the `concatP`

and the `mapP`

vanish due to vectorisation.

## Problem with the naive rules

Nevertheless, these rules are not entirely satisfactory. For example, `[:e | x <- a, b:]`

turns into

`concatMap (\x -> if b then [:e:] else [::]) a`

which is a fairly complicated way to perform

`mapP (\x -> e) . filterP (\x -> b) $ a`

even when taking vectorisation into account. Under vectorisation, the conditional implies `filterP (\x -> b)`

, but adds an expensive, and here useless, merge operation. Maybe these overheads can be optimised away. However, for the moment, we use a desugaring that is based on the above rules, but generates code that should be better suited to array processing.

## Modified rules

The idea is to flatten out the processing of comprehensions to some degree by defining a transformation function `<< . >>`

that gets two arguments: a pattern `pa`

and a desugared expression `ea`

, where we are guaranteed that `ea`

is array valued and all its elements match `pa`

. The semantics of the transformation function is given by

```
<<[: e | qs :]>> pa ea = [: e | pa <- ea, qs :]
= concatMap (\pa -> [: e | qs :]) ea
```

We have the second line by applying Rule (3).

Using this definition of `<< . >>`

, we can derive a new set of desugaring rules. The derivation proceeds by unfold/fold transformations and some properties of the involved combinators. The resulting rules are the following:

```
(1') <<[: e | :]>> pa ea = mapP (\pa -> e) ea
(2') <<[: e | b, qs :]>> pa ea =
<<[: e | qs :]>> pa (filterP (\pa -> b) ea)
(3') <<[: e | p <- a, qs :]>> pa ea =
let ok p = True
ok p = False
in
<<[: q | qs :]>> (pa, p) (crossMapP ea (\pa -> filterP ok a))
(4') <<[: e | let ds, qs :]>> pa ea =
<<[: e | qs :]>> (pa, XS) (mapP (\v@pa -> let ds in (v, XS)) ea)
where XS are the variables bound by ds
(5') <<[: e | qs | qss :]>> pa ea =
<<[: e | qss :]>> (pa, XS) (zipP ea [: XS | qs :])
where XS are the variables bound by qs
```

The typical array processing comprehensions containing only generators, guards, and parallel comprehensions (but not cross-products and lets) are translated into a straight combination of `mapP`

, `filterP`

, and `zipP`

by these rules, which is exactly what we want.