Applications of an odd/even semiring

[GuiltyDolphin]

I recently added a semiring to Granule which has two grades (0 and 1), and has 1 + 1 = 0.

It turns out that this semiring can be used to track whether a hypothesis is discharged an even or odd number of times (see examples/ooz.gr).

Please share any thoughts you have on possible applications for this semiring! I'm imagining that this could have some interesting applications in binary trees (balanced or otherwise).

Examples welcome :)

I am fairly confident that this semiring can be generalised, to form a family of semirings P k for k in Nat>0. Then the operations in the semiring are mod k, thus (n + m)_{P k} = ((n + m) % k)_{Nat} (exploring this over on #149).