ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

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Finite partial sums of a real family form a net directed by inclusion

Example

For a family $(a_{i})_{i \in I}$ of real numbers, let $Fin (I)$ be the finite subsets of $I$ , ordered by inclusion, and put $s_{F} = \sum_{i \in F} a_{i}$ . Then $(s_{F})_{F \in Fin (I)}$ is the finite-subset net. The family is summable with sum $s$ when this net converges to $s$ in the usual topology of $R$ .

Facts & Assumptions

Given: A real family $(a_{i})_{i \in I}$ .

[L1]

A finite-index sum is independent of its enumeration (The sum $\sum_{i \in S} a_{i}$ over a finite index set, and its product form).

[L2]

A union of two finite sets is finite, and sums split across disjoint finite sets (The sum rule: a finite disjoint union is finite with $∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣$ and $∣ ⋃_{i \in I} A_{i} ∣ = \sum_{i \in I} ∣ A_{i} ∣$ , and a sum over a finite index set splits along a partition).

Verification

technique · constructive

1.1

$Fin (I)$ is nonempty because it contains $\emptyset$ , and it is directed because $F \cup G$ is a finite upper bound of $F$ and $G$ .

L2construct

1.2

Therefore $F \mapsto s_{F}$ is a net. If $F \subseteq G$ , then $s_{G} = s_{F} + \sum_{i \in G ∖ F} a_{i}$ , so later values add only terms not already counted.

L1L2

2.1

Thus the displayed finite partial sums form the announced net, and its convergence is a definition of unordered summability.

step 1.1step 1.2discharge-construct∎

Depends on

Used by

Assuming countable choice, a real family is summable as a finite-subset net if and only if it has at most countable support and its nonzero terms are absolutely summable; its sum is independent of the enumeration Theorem

Dependency tree · two levels

28 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

Unconditional convergence (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)