
This is a good article!
In ClickHouse, interval arithmetic is applied to index analysis. A sparse index consists of granules, and each granule is an interval of tuples in lexicographic order. This interval is decomposed into a union of hyperrectangles. Conditions such as comparisons, logic operators, and many other functions are evaluated on these hyperrectangles, yielding boolean intervals. Boolean intervals represent ternary logic (always true, always false, can be true or false). Interesting tricks include: applying functions that are monotonic on ranges (for example, the function "day of month" is monotonic as long as the month does not change), calculating function preimages on intervals, and even calculating preimages of nary functions, which is useful for spacefilling curves, such as Morton or Hilbert curves.
Check for more details: https://github.com/ClickHouse/ClickHouse/blob/master/src/Sto...
Or see examples, such as https://adsb.exposed/

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See clpBNR with SWI Prolog for a way to use interval arithmetic in the broader scope of logic programming.
https://github.com/ridgeworks/clpBNR

I took the idea of interval types and decomposed them to an even lower primitive: Inequality types. An interval type is just an intersection of two inequality types. For example `(>0) & (<1)` is the interval `(0, 1)`.
The nice thing about this decomposition is that applying arithmetic "just works" because you just define them for the inequality primitive. For example:
I prototyped this for TypeScript and created a proposal. It does not contain typelevel arithmetic because TS doesn't do that. I'm not entirely convinced myself of that proposal, but maybe someone finds this interesting:
https://github.com/microsoft/TypeScript/issues/43505