TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-02

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.

There exists a language that is not recognizable

Statement

There exists a language that is not recognizable.

Facts & Assumptions

Given: The halting language $H A L T_{T M}$ .

[L1]

The complement of $H A L T_{T M}$ is not recognizable, by The complement of the halting problem is not recognizable.

[L2]

The symbol $H A L T_{T M}$ names a language, by The halting problem $H A L T_{T M}$ .

[L3]

Recognizability is a property of languages, by Decidable and recognizable languages.

Proof

technique · direct

1.1

By [L2], the complement of $H A L T_{T M}$ is itself a language.

L2given

2.1

By [L1], that language is not recognizable, and [L3] identifies this as exactly the existence claim wanted.

L1L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

EECS 376 Course Notes, Part 6: Computability (standard reference, not scraped)