Alphabeta Math
CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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.

Quotienting by a first parameter lowers local dimension by one

Statement

Let (R,m) be a Noetherian local ring of positive dimension, and let (x1,…,xd) be a system of parameters. Then

dim⁡(R/(x1))=d−1.

Facts & Assumptions

Given: A Noetherian local ring (R,m) of positive dimension and a system of parameters (x1,…,xd).

[L1]

By definition, x1 is the first member of a system of parameters (Systems of parameters and parameter ideals).

[L2]

A first parameter lowers the local dimension by exactly one (A first parameter lowers local dimension by exactly one).

Proof

technique · direct
1.1

The tuple (x1,…,xd) satisfies the hypothesis of [L2] by [L1].

L1L2given
2.1

Therefore dim⁡(R/(x1))=d−1.

L2step 1.1∎

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