ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

Building the diagonal machine against a hypothetical acceptance decider

Example

Assume a black-box decider $H$ for $A_{T M}$ . The diagonal lemma builds a machine $D_{H}$ with input alphabet $Σ_{2} = {1, 2}$ such that on the recoded description $\overline{⌜ M ⌝}$ of any machine $M$ with the same input alphabet, $D_{H}$ does the opposite of $M$ .

Facts & Assumptions

Given: A hypothetical decider $H$ for $A_{T M}$ and the resulting machine $D_{H}$ .

[L1]

The diagonal construction yields a machine $D_{H}$ with $D_{H} accepts \overline{⌜ M ⌝} ⟺ M does not accept \overline{⌜ M ⌝}$ for every coded machine $M$ whose input alphabet contains $Σ_{2}$ , by A hypothetical decider for $A_{T M}$ yields a diagonal self-application machine.

[L2]

The existence of such a machine forces a contradiction and therefore proves $A_{T M}$ undecidable, by The Turing-machine acceptance problem is undecidable.

Verification

technique · direct

1.1

Applying [L1] with $M = D_{H}$ itself gives $D_{H} accepts \overline{⌜ D_{H} ⌝} ⟺ D_{H} does not accept \overline{⌜ D_{H} ⌝}$ . So the special input is the recoded description $\overline{⌜ D_{H} ⌝}$ , not an informal self-reference slogan.

L1given

2.1

The contradiction in step 1.1 is exactly the obstruction recorded in [L2]. This concrete trace is the diagonal core of the undecidability proof.

L2step 1.1∎

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

A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem (standard reference, not scraped)