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Joined 2026-09-21 Message
Incoming links: Occurrence-position representation of a sequence
Use arbitrary-precision integers for positions, avoiding overflow on infinite sequences:The type synonym
type Seq a = a -> [Integer]
rep :: Eq a => [a] -> Seq a
rep xs v = [i | (i,x) <- zip [0..] xs, x == v]Seq a needs no equality constraint; the equality test is required specifically by rep. The list comprehension visits input positions in increasing order, so the output is an increasing list of precisely the matching positions. For finite inputs this includes a terminating empty list when the value is absent. Infinite inputs require the additional productivity distinctions discussed below. This is an occurrence-position representation of a sequence.The most informative occurrence-position representation of a sequence of natural numbers isEach nonnegative integer occurs once at its own index, and negative integers never occur.
naturals :: Seq Integer
naturals v
| v < 0 = []
| otherwise = [v]This is not equal to
rep [0..] under lazy evaluation. For a nonnegative query v, rep eventually emits v, but then searches forever for another occurrence: its result has the shape v : undefined, rather than the terminating singleton [v]. For a negative query, even the outer constructor is never produced, whereas naturals immediately returns [].No general, correct
rep :: Eq a => [a] -> Seq a can always certify those absences from an arbitrary infinite input. A terminating certification has inspected only finitely many elements, and the same prefix could be followed by the supposedly absent value, or by a further occurrence of it. Extra knowledge changes the problem: for a promised strictly increasing integer input, a search can stop with [] when it first exceeds the query and return [i] immediately upon equality. That specialised representation can produce naturals; a generic search cannot. This is the absence-detection obstruction for infinite lists.