ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

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 $\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert$ and $\lvert\bigcup_{i \in I} A_i\rvert = \sum_{i \in I}\lvert A_i\rvert$ , and a sum over a finite index set splits along a partition).

Verification

technique · constructive

1.1

$\operatorname{Fin}(I)$ is nonempty because it contains $\varnothing$ , 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 $s_G=s_F+\sum_{i\in G\setminus F}a_i$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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