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

Nonpermutable Regular Sequence

Example

In $R = k [x, y, z] / ((x - 1) z)$ , $x, (x - 1) y$ is regular, whereas the reverse order is not: $(x - 1) y z = 0$ with $z \neq 0$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Regular Sequence On A Module, Regularity Notions And Permutation Invariance Local.

Proof

technique · direct

1.1givenalgebra

Modulo $x$ , the relation becomes $z = 0$ , leaving $k [y]$ where $(x - 1) y$ becomes $- y$ ; also $x$ is a non-zero-divisor.

2.1step 1.1algebra∎

In reverse order $z \neq 0$ is killed by $x - 1$ , so the first regularity condition fails; the ring is not local.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)