CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Infinitely many pairwise distinguishable prefixes force nonregularity

Statement

Let $L \subseteq Σ^{*}$ . If there are infinitely many words in $Σ^{*}$ that are pairwise distinguished by suffixes, then $L$ is not regular.

Facts & Assumptions

Given: A language $L \subseteq Σ^{*}$ with infinitely many pairwise distinguishable words.

[L1]

By Distinguishing words for states and for prefixes, a suffix distinguishes two words exactly when those two words are not Nerode-equivalent.

[L2]

By A language is regular if and only if its Nerode equivalence has finite index, a regular language has only finitely many Nerode classes.

Proof

technique · direct

1.1

If two words are pairwise distinguished by suffixes, then [L1] shows that they lie in distinct Nerode classes.

L1given

2.1

Therefore the infinitely many given words occupy infinitely many Nerode classes. By [L2], $L$ cannot be regular.

L2step 1.1∎

Depends on

Used by

A nonregular language can still satisfy the regular pumping lemma Counterexample
The prefixes 0ⁿ distinguish the language 0ⁿ1ⁿ : n >= 0 Example
FALSE: the pumping lemma characterizes the regular languages False statement

Dependency tree · two levels

9 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

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
Eric Blais, Models of Computation, 20. Nonregular Languages (standard reference, not scraped)