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.

The compositions of $4$ from ${SEQ}_{\geq 1} ({SEQ}_{\geq 1} (Z))$

Example

The eight compositions of $4$ are

$4,$

$3 + 1, 1 + 3, 2 + 2,$

$2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2,$

$1 + 1 + 1 + 1.$

They split by number of parts as $1, 3, 3, 1$ .

Facts & Assumptions

Given: Compositions are nonempty sequences of positive integers (Positive-integer compositions have generating function $x / (1 - 2 x)$ ).

[L1]

The number of compositions of $n$ into exactly $k$ positive parts is $(\binom{n - 1}{k - 1})$ (Compositions of $n$ into $k$ positive parts are counted by $(\binom{n - 1}{k - 1})$ ).

Verification

technique · direct

1.1

The displayed list contains every ordered positive-part sum of $4$ : one with $1$ part, three with $2$ parts, three with $3$ parts, and one with $4$ parts.

given

2.1

The part-counts match [L1]: $(\binom{3}{0}) = 1$ , $(\binom{3}{1}) = 3$ , $(\binom{3}{2}) = 3$ , and $(\binom{3}{3}) = 1$ .

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.

The compositions of 44 from SEQ⁡≥1(SEQ⁡≥1(Z))SEQ≥1​(SEQ≥1​(Z))