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.

Recognizable languages are closed under union and intersection

Statement

If $L, K \subseteq Σ^{*}$ are recognizable, then both $L \cup K$ and $L \cap K$ are recognizable.

Facts & Assumptions

Given: Recognizable languages $L, K \subseteq Σ^{*}$ .

[L1]

By Decidable and recognizable languages, a recognizer accepts exactly the members of its language, but may diverge on nonmembers.

[L2]

By Boolean operations on languages over a fixed alphabet, $L \cup K$ and $L \cap K$ are languages over the same alphabet.

Proof

technique · direct

1.1

Let $M_{L}$ recognize $L$ and $M_{K}$ recognize $K$ . To recognize $L \cup K$ , interleave their computations on an input $w$ and accept as soon as either machine accepts. If $w \in L \cup K$ , one of the recognizers eventually accepts; if $w \notin L \cup K$ , neither does. Hence $L \cup K$ is recognizable by [L1] and [L2].

L1L2given

1.2

To recognize $L \cap K$ , again interleave the two computations on $w$ , but now record whether each machine has accepted and halt only once both accept. If $w \in L \cap K$ , both accepting events occur after finitely many stages; if $w \notin L \cap K$ , at least one machine never accepts, which is allowed for a recognizer. Thus $L \cap K$ is recognizable.

L1L2construct

2.1

Therefore recognizable languages are closed under union and intersection.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

John Watrous, Introduction to the Theory of Computing, Lecture 18: Further discussion of computability (standard reference, not scraped)
Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)