ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Localizing a Dedekind domain at a nonzero prime

Example

Let $R$ be a Dedekind domain, let $p$ be a nonzero prime ideal, and let $n \geq 0$ be an integer. Then $R_{p}$ is a DVR, and

(p^{n})_{p} = p^{n} R_{p} .

Hence $v_{p} (p^{n}) = n$ .

Facts & Assumptions

Given: A Dedekind domain $R$ , a nonzero prime ideal $p$ , and an integer $n \geq 0$ .

[L1]

The localisation $R_{p}$ is a discrete valuation ring (Localizing a Dedekind domain at a nonzero prime gives a DVR).

[F1]

The valuation $v_{p} (I)$ is defined by the equality $I_{p} = p^{v_{p} (I)} R_{p}$ (Prime-ideal valuations on fractional ideals).

Verification

technique · direct

1.1

By [L1], the localisation $R_{p}$ is a DVR. Localising the ideal $p^{n}$ gives exactly $p^{n} R_{p}$ by the definition of localisation of ideals.

L1given

2.1

Comparing step 1.1 with [F1] shows that the corresponding valuation is $v_{p} (p^{n}) = n$ .

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

J. P. May, Notes on Dedekind Rings (standard reference, not scraped)
Mircea Mustata, Introduction to Commutative Algebra, §8.5 (standard reference, not scraped)