ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Trivial factors in an elementary group

Example

For every prime $p$ , every finite $p$ -group is $p$ -elementary by taking $C = 1$ ; every cyclic group of order prime to $p$ is $p$ -elementary by taking $P = 1$ ; and $1 = 1 \times 1$ is $p$ -elementary and $p$ -hyperelementary.

Facts & Assumptions

[F1]

The cited prerequisite is $p$ -elementary and $p$ -hyperelementary finite groups.

Verification

Given: the trivial group is cyclic and has order $1$ .

1.1

The order $1$ is prime to every prime, and it is also $p^{0}$ .

F1given

2.1

Therefore each displayed choice satisfies both factor conditions in the definition, including the simultaneous trivial-factor case. ∎

step 1.1

Depends on

$p$-elementary and $p$-hyperelementary finite groups

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Wen-Wei Li, Yanqi Lake Lectures on Algebra I, Definition 14.1.1 (standard reference, not scraped)