TheoremStatement: 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.

A language is regular if and only if some regular expression denotes it

Statement

Let $L \subseteq Σ^{*}$ . Then $L$ is regular if and only if there exists a regular expression $R$ over $Σ$ with $L (R) = L .$

Facts & Assumptions

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

[L1]

By Regular languages, a language is regular exactly when some DFA recognizes it.

[L2]

By Every regular expression has an equivalent epsilon-NFA, every regular expression has an equivalent epsilon-NFA.

[L3]

By The subset construction preserves the recognized language, the subset construction turns an epsilon-NFA into a DFA with the same recognized language.

[L4]

By Every DFA language is denoted by a regular expression, every DFA language is denoted by some regular expression.

Proof

technique · direct

1.1

Suppose first that $L (R) = L$ for some regular expression $R$ . By [L2], there is an epsilon-NFA recognizing $L$ , and by [L3] there is then a DFA recognizing the same language. Therefore [L1] implies that $L$ is regular.

L1L2L3given

1.2

Conversely, suppose that $L$ is regular. By [L1], some DFA $D$ recognizes $L$ . Then [L4] gives a regular expression $R$ with $L (R) = L (D) = L$ .

L1L4given

2.1

Steps 1.1 and 1.2 prove the forward and reverse implications.

step 1.1step 1.2∎

Depends on

Used by

Regular languages are closed under homomorphic image Theorem

Dependency tree · two levels

15 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)