ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge 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.

First three chacon tower heights

Example

The first three formal Chacon tower heights are $h_{0} = 1$ , $h_{1} = 4$ , $h_{2} = 13$ . For every integer $r \geq 0$ , $h_{r} = (3^{r + 1} - 1) / 2$ .

Facts & Assumptions

[F1]

The finite tower recursion is $h_{0} = 1$ , $h_{r + 1} = 3 h_{r} + 1$ Chacon three cut one spacer towers. Its finite combinatorial clause is choice-free.

Verification

Given: The formal recursion of F1.

1.1F1

Direct substitution gives $h_{1} = 3 \cdot 1 + 1 = 4$ and $h_{2} = 3 \cdot 4 + 1 = 13$ . The extra one counts the spacer level, whereas the three copies count the old levels in the three columns.

2.1F1step 1.1∎

Set $b_{r} = (3^{r + 1} - 1) / 2$ . Then $b_{0} = (3 - 1) / 2 = 1$ and $3 b_{r} + 1 = (3^{r + 2} - 3 + 2) / 2 = (3^{r + 2} - 1) / 2 = b_{r + 1}$ . Induction on $r$ using the same initial value and recurrence proves $h_{r} = b_{r}$ . At $r = 1, 2$ this gives $(9 - 1) / 2 = 4$ and $(27 - 1) / 2 = 13$ , agreeing with step 1.1. Only finite arithmetic and induction are used; no measure existence or choice assumption is used.

Depends on

Chacon three cut one spacer towers

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

Sarig Problem 3.8 pp.99–101 (standard reference, not scraped)