DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Martin-Löf tests and random sequences

Definition

A sequence $(U_{n})_{n \geq 0}$ is uniformly effectively open when there is one computably enumerable relation $W \subseteq N \times {0, 1}^{*}$ such that $U_{n} = ⋃_{(n, σ) \in W} [σ]$ for every $n$ . Thus a single algorithm, given no oracle, enumerates all pairs $(n, σ)$ and thereby enumerates the cylinders at every level uniformly.

A Martin-Löf test is a uniformly effectively open sequence with $μ (U_{n}) \leq 2^{- n}$ . A sequence $X \in 2^{ω}$ is Martin-Löf random when $X \notin ⋂_{n} U_{n}$ for every such test. Computable enumerability, cylinders, and measure are as in Effectively open sets in Cantor space.

Depends on

Effectively open sets in Cantor space

Used by

False: normality implies algorithmic randomness False statement
A universal Martin-Löf test exists Lemma
Levin--Schnorr characterization of Martin-Löf randomness Theorem
Martin-Löf randomness implies computable randomness Theorem

Dependency tree · two levels

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Sources

Simpson, §8.2 (standard reference, not scraped)