ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

dvrs as regular local rings

Example

A DVR $R$ with uniformizer $t$ and residue field $k$ is regular local of dimension one, with regular system $(t)$ and ${gr}_{(t)} R ≅ k [T]$ .

Facts & Assumptions

Given: The objects and hypotheses in the example. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

one dimensional regular local rings are dvrs: A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring. Fields are excluded from the term DVR.

[F2]

associated graded ring of a regular local ring: If $(R, m, k)$ is regular local of dimension $d$ , any cotangent basis induces a graded isomorphism $k [X_{1}, \dots, X_{d}] ≅ {gr}_{m} R$ . Conversely, if the associated graded ring of a nonzero Noetherian local ring is isomorphic as a graded $k$ -algebra to $k [X_{1}, \dots, X_{d}]$ with standard grading, then $R$ is regular of dimension $d$ .

Verification

1.1

The DVR equivalence gives dimension-one regularity. Its maximal ideal is $(t)$ and $t \notin (t^{2})$ , since otherwise cancellation would make the nonunit $t$ a unit. Thus its cotangent basis is the class of $t$ .

F1algebra

2.1

For every $n \geq 0$ , multiplication by $t^{n}$ identifies $k$ with $(t^{n}) / (t^{n + 1})$ : injectivity follows by cancellation and surjectivity by principality. Products of these classes are powers of the degree-one class, so the graded map $k [T] \to {gr}_{(t)} R$ is an isomorphism, also as given by the regular graded theorem.

F2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Example 12.10, p.116 (standard reference, not scraped)