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.

Epsilon-closure is a closure operator

Statement

Let $N$ be an NFA with epsilon-moves and let $S, T$ be subsets of its state set. Then:

$S \subseteq ECl (S)$ ;
if $S \subseteq T$ , then $ECl (S) \subseteq ECl (T)$ ;
$ECl (ECl (S)) = ECl (S)$ .

Facts & Assumptions

Given: An NFA with epsilon-moves $N$ and subsets $S, T$ of its state set.

[L1]

By The epsilon-closure of a set of NFA states, a state lies in $ECl (S)$ exactly when it is reachable from some state of $S$ by a finite chain of $ε$ -transitions.

Proof

technique · direct

1.1

For every $s \in S$ , the length-zero $ε$ -chain from $s$ to itself shows $s \in ECl (S)$ by [L1]. Hence $S \subseteq ECl (S)$ .

L1given

1.2

Assume $S \subseteq T$ and let $q \in ECl (S)$ . By [L1], some $ε$ -chain starts in a state of $S$ and ends at $q$ ; because that start state also lies in $T$ , the same chain shows $q \in ECl (T)$ . Thus $ECl (S) \subseteq ECl (T)$ .

L1given

2.1

Step 1.1 gives $ECl (S) \subseteq ECl (ECl (S))$ . For the reverse inclusion, let $q \in ECl (ECl (S))$ . By [L1], there is an $ε$ -chain from some $r \in ECl (S)$ to $q$ , and again by [L1] there is an $ε$ -chain from some $s \in S$ to $r$ . Concatenating the two finite chains gives an $ε$ -chain from $s$ to $q$ , so $q \in ECl (S)$ . Therefore $ECl (ECl (S)) = ECl (S)$ .

L1step 1.1∎

Depends on

The epsilon-closure of a set of NFA states

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)
John Watrous, Introduction to the Theory of Computing, Lecture 3: Nondeterministic finite automata (standard reference, not scraped)