ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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 finite p-group is naturally isomorphic to its own pro-p completion

Example

If $P$ is a finite $p$ -group, then the canonical map $P \to {\hat{P}}^{(p)}$ is an isomorphism.

Facts & Assumptions

Given: A finite $p$ -group $P$ .

[F1]

The pro- $p$ completion is the inverse limit over all finite $p$ -group quotients (The pro-p completion of an abstract group is the inverse limit over its finite p-group quotients).

[L1]

A finite $p$ -group is itself a pro- $p$ group (A pro-p group is a profinite group that is an inverse limit of finite p-groups).

Verification

technique · direct

1.1

The subgroup $1 ⊴ P$ is one of the indexing subgroups in [F1], and its quotient is the initial object $P / 1 ≅ P$ of the quotient diagram: it has the natural quotient map to every $P / N$ .

F1givenalgebra

2.1

A compatible tuple is uniquely determined by its coordinate in the initial object $P / 1$ , and every element of $P / 1$ determines such a tuple by its images in the other quotients. Therefore ${\hat{P}}^{(p)} ≅ P$ , and the canonical completion map is exactly this isomorphism. The conclusion is consistent with [L1], since $P$ was already pro- $p$ to begin with.

L1step 1.1∎

Depends on

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

Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)