Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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.

Constructing two generators for a Dedekind ideal

Example

Let I be a nonzero ideal in a Dedekind domain R, and choose 0≠a∈I. Then the proof of the two-generator theorem constructs an element b∈I such that I=(a,b) by correcting only the finitely many prime valuations at which (a) is too large.

Facts & Assumptions

Given: A Dedekind domain R, a nonzero ideal I, and a chosen nonzero element a∈I.

[L1]

Every nonzero ideal of a Dedekind domain is generated by the chosen element a together with one further element b∈I (Every nonzero ideal in a Dedekind domain is generated by two elements).

Verification

technique · direct
1.1

The theorem [L1] identifies a finite set of bad primes, namely those for which vp(a)>vp(I), chooses local correction terms at those primes, and combines them by the Chinese remainder step in its proof.

L1given
2.1

The resulting element b∈I has exactly the missing prime valuations, so (a,b)=I. This is the concrete content of the two-generator construction.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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