LemmaStatement: 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.

Nerode-equivalent words stay equivalent after right concatenation

Statement

For every language $L \subseteq Σ^{*}$ , the Nerode relation $\equiv_{L}$ of Nerode equivalence of words is an equivalence relation on $Σ^{*}$ , and for every words $u, v, x \in Σ^{*}$ , $u \equiv_{L} v ⟹ u x \equiv_{L} v x .$

Facts & Assumptions

Given: A language $L \subseteq Σ^{*}$ .

[L1]

By Nerode equivalence of words, $u \equiv_{L} v$ means that for every continuation $y \in Σ^{*}$ one has $u y \in L$ if and only if $v y \in L$ .

[L2]

By Equivalence relation, equivalence class, and the quotient set $A / \sim$ , it is enough to check reflexivity, symmetry, and transitivity.

Proof

technique · direct

1.1

Reflexivity, symmetry, and transitivity follow immediately from the biconditional in [L1], so [L2] shows that $\equiv_{L}$ is an equivalence relation.

L1L2given

1.2

Assume $u \equiv_{L} v$ and fix $x \in Σ^{*}$ . For every continuation $y$ , [L1] applied to the continuation $x y$ gives $u x y \in L ⟺ v x y \in L$ , which is exactly the condition $(u x) \equiv_{L} (v x)$ .

L1given

2.1

Therefore $\equiv_{L}$ is an equivalence relation preserved by right concatenation.

step 1.1step 1.2∎

Depends on

Used by

A language is regular if and only if its Nerode equivalence has finite index Theorem

Dependency tree · two levels

11 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)
Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)