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.

Partitions with parts at most $3$ from a truncated multiset product

Example

Restricting the Euler product to part sizes $1, 2, 3$ gives

$\frac{1}{(1 - x) (1 - x^{2}) (1 - x^{3})} .$

The coefficient of $x^{5}$ is $5$ , corresponding to the five partitions of $5$ whose parts are at most $3$ :

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

Facts & Assumptions

Given: A combinatorial class having one object of each size $1$ , $2$ , and $3$ , and no other objects.

[L1]

If $A$ has no size-zero objects and $a_{n}$ objects of size $n$ , then $MSET (A)$ has generating function $\prod_{n \geq 1} (1 - x^{n})^{- a_{n}}$ (If $A$ has no size-zero objects then $MSET (A)$ has generating function $\prod_{n \geq 1} (1 - x^{n})^{- a_{n}}$ ).

Verification

technique · direct

1.1

Applying [L1] to the Given class and multiplying the resulting factors to degree $5$ gives $(1 - x)^{- 1} (1 - x^{2})^{- 1} (1 - x^{3})^{- 1} = (1 + x + x^{2} + x^{3} + x^{4} + x^{5}) (1 + x^{2} + x^{4}) (1 + x^{3}) = 1 + x + 2 x^{2} + 3 x^{3} + 4 x^{4} + 5 x^{5} + O (x^{6})$ .

givenL1algebra

2.1

The five displayed partitions of $5$ are exactly the partitions whose part sizes lie in ${1, 2, 3}$ , so the coefficient $5$ has the advertised interpretation.

step 1.1∎

Depends on

If $\mathcal{A}$ has no size-zero objects then $\operatorname{MSET}(\mathcal{A})$ has generating function $\prod_{n\ge 1}(1-x^n)^{-a_n}$

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.

Partitions with parts at most 33 from a truncated multiset product