ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

A parameter sequence that fails in a non-Cohen--Macaulay ring

Example

In the one-dimensional local ring $A = k ⟦ x, y ⟧ / (x^{2}, x y)$ , the one-element sequence $y$ is a system of parameters but is not regular.

Facts & Assumptions

Given: ex-non-cohen-macaulay-local-ring computes $\dim A = 1$ and $depth A = 0$ .

Verification

technique · direct

1.1given

The quotient $A / y A ≅ k ⟦ x ⟧ / (x^{2})$ has dimension $0$ , so $y$ is a parameter.

2.1step 1.1algebra∎

But the nonzero class of $x$ satisfies $y x = 0$ . Thus $y$ is a zero divisor and the parameter sequence is not regular, matching the depth gap $0 < 1$ .

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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)