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

1.1

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 $q_{0}$ .

F1given

2.1

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.

F1step 1.1

3.1

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.

F1step 2.1∎

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)