# Couple and Merge

Solo/Couple (`≍`) and Merge (`>`) are functions that build a higher-rank array out of cells. Each takes some input arrays with the same shape, and combines them into a single array that includes all their elements. For example, let's couple two arrays of shape `2‿3`:

↗️
```    ⟨p←3‿5×⌜↕3, q←2‿3⥊"abcdef"⟩  # Shown side by side
┌─
· ┌─         ┌─
╵ 0 3  6   ╵"abc
0 5 10     def"
┘       ┘
┘

p ≍ q   # p coupled to q
┌─
╎ 0   3   6
0   5   10

'a' 'b' 'c'
'd' 'e' 'f'
┘

≢ p ≍ q
⟨ 2 2 3 ⟩
```

The result is an array with `p` and `q` for its major cells. It has two inner axes that are shared by `p` and `q`, preceded by an outer axis, with length 2 because there are two arguments. With no left argument, `≍` does something simpler: it just adds an axis of length 1 to the front. The argument goes solo, becoming the only major cell of the result.

↗️
```    ≍ q
┌─
╎"abc
def"
┘

≢ ≍ q
⟨ 1 2 3 ⟩
```

Merge (`>`) takes one argument, but a nested one. `𝕩` is an array of arrays, each with the same shape. The shape of the result is then the outer shape `≢𝕩` followed by this shared inner shape.

↗️
```    ⊢ a ← "AB"‿"CD" ∾⌜ "rst"‿"uvw"‿"xyz"
┌─
╵ "ABrst" "ABuvw" "ABxyz"
"CDrst" "CDuvw" "CDxyz"
┘

> a
┌─
╎"ABrst
ABuvw
ABxyz

·CDrst
CDuvw
CDxyz"
┘

≢ a
⟨ 2 3 ⟩
≢ > a
⟨ 2 3 5 ⟩
```

If `𝕩` is empty, but has a fill element, then its shape is used for the inner shape. If it doesn't have a fill, the inner shape is assumed to be empty, so that the result is `𝕩` with no changes.

Merge serves as a generalization of Solo and Couple, since Solo is `{>⟨𝕩⟩}` and Couple is `{>⟨𝕨,𝕩⟩}`. These can be combined with Pair, giving `>⋈` for both. Since the result of `⋈` has rank 1, it can only add one dimension, but `>` can take any number of dimensions as its input.

## Coupling units

A note on the topic of Solo and Couple applied to units. As always, one axis will be added, so that the result is a list (strangely, J's laminate differs from Couple in this one case, as it adds an axis to get a shape `2‿1` result). Solo on a unit is interchangeable with Deshape (`⥊`), and either primitive might be chosen for stylistic reasons. Couple on units is equivalent to Join-to (`∾`), but this is an irregular form of Join-to because it is the only case where Join-to adds an axis to both arguments instead of just one. Couple should be preferred in this case.

As a consequence, Pair (`⋈`) can be written `≍○<`, while `≍` is `>∘⋈` as discussed above. This gives the neat (but not useful) identities `≍ ←→ >∘≍○<`, and `⋈ ←→ >∘⋈○<`, which have the same form because adding `○<` commutes with adding `>∘`.

## Merge and array theory

In all cases, what these functions do is more like reinterpreting existing data than creating new information. In fact, if we ignore the shape and look at the deshaped arrays involved in a call to Merge, we find that it just joins them together. Essentially, Merge is a request to ensure that the inner arrays make up a homogeneous (not "ragged") array, and then to consider them to be such an array. It's the same thing Rank does to combine the result cells from its operand into a single array.

↗️
```    ⥊ > a
"ABrstABuvwABxyzCDrstCDuvwCDxyz"

⥊ ⥊¨ a
⟨ "ABrst" "ABuvw" "ABxyz" "CDrst" "CDuvw" "CDxyz" ⟩
∾ ⥊ ⥊¨ a
"ABrstABuvwABxyzCDrstCDuvwCDxyz"
```

Somewhat like Table `⌜`, Merge might be considered a fundamental way to build up multidimensional arrays from lists. In both cases rank-0 or unit arrays are somewhat special. They are the identity value of a function with Table, and Enclose (`<`), which creates a unit, is a right inverse to Merge. Enclose is needed because Merge can't produce a rank 0 array on its own. Merge has another catch as well: it can't produce arrays with a 0 in the shape, except at the end, without relying on a fill element.

↗️
```    ⊢ e ← ⟨⟩¨ ↕3
⟨ ⟨⟩ ⟨⟩ ⟨⟩ ⟩
≢ > e
⟨ 3 0 ⟩
≢ > > e
⟨ 3 0 ⟩
```

Above we start with a list of three empty arrays. After merging once we get a shape `3‿0` array, sure, but what happens next? The shape added by another merge is the shared shape of that array's elements—and there aren't any! If the nested list kept some type information around then we might know, but extra type information is essentially how lists pretend to be arrays. True dynamic lists simply can't represent multidimensional arrays with a 0 in the middle of the shape. In this sense, arrays are a richer model than nested lists.

## Definitions

We can define `≍` as `>⋈`. To complete the picture we should describe Merge fully. Merge is defined on an array argument `𝕩` such that there's some shape `s` satisfying `∧´⥊(s≡≢)¨𝕩`. If `𝕩` is empty then any shape satisfies this expression; `s` is then the shape of the fill if there is one, or otherwise `⟨⟩`.

Then the result of Merge is `𝕩⊑˜⌜↕s`. Here, Table is a nice way of combining outer and inner axes to produce the result; since the outer axes come first they should go on the left and the inner axes on the right. `𝕩` is a natural choice of left argument, and because no concrete array can be used, the right argument is `↕s`, the array of indices into any element of `𝕩`. Then Pick selects all the elements. If `s` is empty, then `𝕩` is allowed to contain atoms as well as unit arrays. Pick will implicitly treat them as arrays.