# Module Diffing

    module Diffing: sig .. end

# Parametric diffing

This module implements diffing over lists of arbitrary content. It is parameterized by

- The content of the two lists
- The equality witness when an element is kept
- The diffing witness when an element is changed

Diffing is extended to maintain state depending on the computed changes while walking through the two lists.

The underlying algorithm is a modified Wagner-Fischer algorithm (see <https://en.wikipedia.org/wiki/Wagner%E2%80%93Fischer_algorithm>).

We provide the following guarantee: Given two lists `l` and `r`, if different patches result in different states, we say that the state diverges.

- We always return the optimal patch on prefixes of `l` and `r` on which state does not diverge.
- Otherwise, we return a correct but non-optimal patch where subpatches with no divergent states are optimal for the given initial state.

More precisely, the optimality of Wagner-Fischer depends on the property that the edit-distance between a k-prefix of the left input and a l-prefix of the right input d(k,l) satisfies

d(k,l) = min ( del_cost + d(k-1,l), insert_cost + d(k,l-1), change_cost + d(k-1,l-1) )

Under this hypothesis, it is optimal to choose greedily the state of the minimal patch transforming the left k-prefix into the right l-prefix as a representative of the states of all possible patches transforming the left k-prefix into the right l-prefix.

If this property is not satisfied, we can still choose greedily a representative state. However, the computed patch is no more guaranteed to be globally optimal. Nevertheless, it is still a correct patch, which is even optimal among all explored patches.

    type ('left, 'right, 'eq, 'diff) change =

`|` | `Delete of `'left``
`|` | `Insert of `'right``
`|` | `Keep of `'left * 'right * 'eq``
`|` | `Change of `'left * 'right * 'diff``

The type of potential changes on a list.

    val map : ('l1 -> 'l2) ->
           ('r1 -> 'r2) ->
           ('l1, 'r1, 'eq, 'diff) change ->
           ('l2, 'r2, 'eq, 'diff) change

    type ('l, 'r, 'eq, 'diff) patch = ('l, 'r, 'eq, 'diff) change list

A patch is an ordered list of changes.

    val diff : weight:(('l, 'r, 'eq, 'diff) change -> int) ->
           test:('state -> 'l -> 'r -> ('eq, 'diff) result) ->
           update:(('l, 'r, 'eq, 'diff) change -> 'state -> 'state) ->
           'state -> 'l array -> 'r array -> ('l, 'r, 'eq, 'diff) patch

`diff ~weight ~test ~update state l r` computes the diff between `l` and `r`, using the initial state `state`.

- `test st xl xr` tests if the elements `xl` and `xr` are compatible (`Ok`) or not (`Error`).
- `weight ch` returns the weight of the change `ch`. Used to find the smallest patch.
- `update ch st` returns the new state after applying a change.

## Variadic diffing

Variadic diffing allows to expand the lists being diffed during diffing.

    type ('l, 'r, 'e, 'd, 'state) update =

`|` | `Without_extensions of `(('l, 'r, 'e, 'd) change -> 'state -> 'state)``
`|` | `With_left_extensions of `(('l, 'r, 'e, 'd) change -> 'state -> 'state * 'l array)``
`|` | `With_right_extensions of `(('l, 'r, 'e, 'd) change -> 'state -> 'state * 'r array)``

    val variadic_diff : weight:(('l, 'r, 'eq, 'diff) change -> int) ->
           test:('state -> 'l -> 'r -> ('eq, 'diff) result) ->
           update:('l, 'r, 'eq, 'diff, 'state) update ->
           'state -> 'l array -> 'r array -> ('l, 'r, 'eq, 'diff) patch

`variadic_diff ~weight ~test ~update state l r` behaves as `diff` with the following difference:

- `update` must now be an `Diffing.update` which indicates in which direction the expansion takes place.