ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Mean and max aggregation can lose multiset multiplicity

Example

Let $d \geq 1$ and $x \in R^{d}$ . For a nonempty finite multiset $S$ of vectors, define its mean as $∣ S ∣^{- 1} \sum_{y \in S} y$ and its coordinatewise maximum to have $j$ -th coordinate $\max_{y \in S} y_{j}$ . The distinct multisets $S_{1} = {⁣ {x} ⁣}$ and $S_{2} = {⁣ {x, x} ⁣}$ have the same mean $x$ and the same coordinatewise maximum $x$ . Thus both mean aggregation and coordinatewise max aggregation can identify different multisets when multiplicities differ.

Example references

finite multisets and multiplicities

neighbor aggregation as a function on multisets

Xu, Hu, Leskovec, and Jegelka, “How Powerful are Graph Neural Networks?”, §5.2 and Appendices G–H

Li and Leskovec, Graph Representation Learning, Chapter 5 §5.3.4

Facts & Assumptions

Given: $d \geq 1$ , a vector $x \in R^{d}$ , and the two finite multisets in the Example. Verify their inequality and aggregator outputs.

[F1]

Finite multisets as finitely supported multiplicity functions: Multiset equality requires equality of each element's multiplicity, and size is total multiplicity.

Verification

1.1givenF1algebra

The multiplicity of $x$ is $1$ in $S_{1}$ and $2$ in $S_{2}$ , so $S_{1} \neq S_{2}$ by [F1].

2.1givenstep 1.1algebra∎

The sums are $x$ and $2 x$ , with sizes $1$ and $2$ , so both means equal $x$ . In each coordinate, the maximum of one copy of $x_{j}$ and the maximum of two equal copies of $x_{j}$ are both $x_{j}$ . Thus both aggregators return the same value on the distinct multisets.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka, How Powerful are Graph Neural Networks? (standard reference, not scraped)
Pan Li and Jure Leskovec, Graph Representation Learning (standard reference, not scraped)