ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Binary words of length at most three from $SEQ (2 Z)$

Example

The class $SEQ (2 Z)$ gives binary words. Up to length $3$ the words are

$ϵ,$

$0, 1,$

$00, 01, 10, 11,$

$000, 001, 010, 011, 100, 101, 110, 111.$

So the initial coefficient sequence is $1, 2, 4, 8$ .

Facts & Assumptions

Given: The binary-word generating function $OGF (B) = 1 / (1 - 2 x)$ (Binary words have generating function $1 / (1 - 2 x)$ ).

Verification

technique · direct

1.1

The displayed list has $1$ word of length $0$ , $2$ of length $1$ , $4$ of length $2$ , and $8$ of length $3$ .

given

2.1

Expanding $1 / (1 - 2 x)$ as $1 + 2 x + 4 x^{2} + 8 x^{3} + \dots$ gives the same initial coefficients, so the explicit list matches the theorem.

step 1.1∎

Depends on

Binary words have generating function $1/(1-2x)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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.

Binary words of length at most three from SEQ⁡(2Z)SEQ(2Z)