ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

A partition into k nonempty blocks generates a sigma-algebra with 2^k members

Example

If $P_{0}, \dots, P_{k - 1}$ are the nonempty blocks of a partition of $X$ , then

$σ_{X} ({P_{0}, \dots, P_{k - 1}}) = {⋃_{i \in S} P_{i} : S \subseteq k}$

and this sigma-algebra has $2^{k}$ members.

Facts & Assumptions

Given: A natural number $k$ and a partition $(P_{i})_{i < k}$ of $X$ into nonempty blocks.

[L1]

A countable partition generates exactly the unions of its blocks, and the subset-to-union map is a bijection (A countable partition generates exactly the unions of its blocks, and the resulting sigma-algebra is countable exactly for a finite partition).

Verification

technique · direct

1.1L1

Applying [L1] to the finite index set $k$ gives the displayed sigma-algebra and a bijection from $P (k)$ to it.

2.1step 1.1algebra∎

The finite power set $P (k)$ has $2^{k}$ members. For $k = 0$ , necessarily $X = \emptyset$ and there is one union; for $k = 1$ , the two unions are $\emptyset$ and $X$ .

Depends on

A countable partition generates exactly the unions of its blocks, and the resulting sigma-algebra is countable exactly for a finite partition

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Examples 2.4-2.6 (standard reference, not scraped)