Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)
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.

Depth of a hypersurface quotient

Example

Assume the Axiom of Choice (The Axiom of Choice). Let (Q,n) be a regular local domain of dimension d>0 and let 0≠f∈n. Then depth⁡(Q/(f))=d−1=dim⁡(Q/(f)).

Facts & Assumptions

Given: The Axiom of Choice; Q is Cohen--Macaulay of depth d, and f is a nonzerodivisor.

Verification

technique · direct
1.1given

The depth quotient formula gives depth⁡(Q/(f))=depth⁡(Q)−1=d−1.

2.1step 1.1algebra∎

The principal ideal theorem and regularity of f give dim⁡(Q/(f))=d−1. Thus the hypersurface quotient is Cohen--Macaulay.

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