ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Boone halting cleanup through an empty right tape

Example

For any tape symbols $a, b \in S$ , including $a = b = s_{0}$ , the augmented stopping configuration $h a q_{0} b h$ has the cleanup $h a q_{0} b h ⟶ h a q_{0} h ⟶ h q_{0} h ⟶ q .$ The middle two words have empty right tape.

Facts & Assumptions

Given: Tape symbols $a, b \in S$ and the displayed stopping configuration. No reachability from a particular input is claimed.

[F1]

Cleanup relations are $q_{0} s = q_{0}$ , $s q_{0} h = q_{0} h$ for $s \in S$ , and $h q_{0} h = q$ . Stopping configurations may have empty right tape. (Boone machine semigroup and augmented configurations)

Verification

1.1F1given

Substitute $q_{0} b = q_{0}$ with prefix $h a$ and suffix $h$ . The result is $h a q_{0} h$ , whose left tape is the one-letter word $a$ and whose right tape is empty. It is augmented because its state is $q_{0}$ .

2.1F1step 1.1

Substitute $a q_{0} h = q_{0} h$ with prefix $h$ . The result is $h q_{0} h$ , with both tape contexts empty and the same stopping state.

3.1F1step 2.1∎

Substitute $h q_{0} h = q$ with both contexts empty. This produces the terminal augmented word $q$ . Each preceding word had one state and the two outer markers; this last rule removes the markers. The three substitutions remain valid for $a = b = s_{0}$ , giving a concrete all-blank instance. Thus requiring nonempty right tape throughout cleanup would exclude the explicitly reached middle words.

Depends on

Boone machine semigroup and augmented configurations

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

Rotman, Chapter 12, p.427, cleanup relations (standard reference, not scraped)