Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passverified 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.

Regular Sequence Powers And Permutation

Example

Assume the Axiom of Choice. In k[x,y](x,y), (x,y), (x2,y3), and (y3,x2) are regular sequences.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and a field k. The declared prerequisites used here are Positive Powers Of A Regular Sequence Remain Regular and Regular Sequences Permutable Local.

Proof

technique · direct
1.1givenalgebra

x,y are regular in the local polynomial ring because the successive quotients are domains.

2.1step 1.1algebra∎

Under the assumed AC, Positive Powers Of A Regular Sequence Remain Regular applied to (x,y) gives (x2,y3); Regular Sequences Permutable Local then swaps its two entries to give (y3,x2). These two applications are the uses of Choice.

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