ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-12

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.

The prefixes 0^n distinguish the language {0^n1^n : n >= 0}

Example

Let $L : = {0^{n} 1^{n} : n \geq 0} .$ The prefixes $0^{n}$ give an infinite family of pairwise distinguishable words for $L$ .

Facts & Assumptions

Given: The language $L = {0^{n} 1^{n} : n \geq 0}$ .

[L1]

By Distinguishing words for states and for prefixes, a suffix distinguishes two prefixes exactly when appending it puts one resulting word in $L$ and the other outside $L$ .

[L2]

By Infinitely many pairwise distinguishable prefixes force nonregularity, infinitely many pairwise distinguishable prefixes force nonregularity.

Verification

technique · direct

1.1

Fix integers $i < j$ . Appending the suffix $1^{i}$ gives $0^{i} 1^{i} \in L$ and $0^{j} 1^{i} \notin L$ , so [L1] shows that $1^{i}$ distinguishes the prefixes $0^{i}$ and $0^{j}$ .

L1given

2.1

Because every pair $0^{i}, 0^{j}$ with $i < j$ is distinguished by step 1.1, the family ${0^{n} : n \geq 0}$ is pairwise distinguishable. Therefore [L2] implies that $L$ is not regular.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eric Blais, Models of Computation, 20. Nonregular Languages (standard reference, not scraped)