LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A keyed permutation stream does not reuse outputs before its counter interval ends

Statement

For a fixed key and counter interval in Counter stream from a keyed finite permutation, distinct counters produce distinct output words. If the interval is the full set ${0, \dots, 2^{w} - 1}$ , every word of that finite state set appears exactly once. These are deterministic no-reuse statements; they imply no probabilistic independence.

Facts & Assumptions

Given: A fixed key, a bijection $π_{k} : S \to S$ , and the counter procedure of Counter stream from a keyed finite permutation.

[F1]

For a fixed key, the supplied map $π_{k}$ is a bijection of the finite counter set. Counter stream from a keyed finite permutation

Proof

1.1givenF1

If $c \neq c^{'}$ are counters in the interval, injectivity of $π_{k}$ gives $π_{k} (c) \neq π_{k} (c^{'})$ . The procedure uses each counter at most once before returning exhausted, so no output repeats in that interval.

2.1F1step 1.1algebra∎

On the full counter set, the injective map $π_{k}$ has $2^{w}$ distinct outputs in a set of size $2^{w}$ , hence its range is all of $S$ . Thus every word appears exactly once, with no probability assertion.

Depends on

Counter stream from a keyed finite permutation

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources

John K. Salmon et al., Parallel Random Numbers: As Easy as 1, 2, 3, §2 (standard reference, not scraped)