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.

Every nonfield PID is a Dedekind domain with trivial class group

Example

Assume the Axiom of Choice. Every principal ideal domain that is not a field is a Dedekind domain, and its ideal class group is trivial.

Facts & Assumptions

Given: A principal ideal domain R that is not a field.

[F1]

In a principal ideal domain every ideal is principal (Principal ideal domain).

[L1]

Every principal ideal domain is Noetherian (Every principal ideal domain is Noetherian).

[L2]

A nonfield Noetherian domain is Dedekind exactly when every nonzero proper ideal is locally principal (Equivalent local characterizations of Dedekind domains).

[L3]

A Dedekind domain is a PID exactly when its class group is trivial (A Dedekind domain is a PID exactly when its class group is trivial).

Verification

technique · direct
1.1

By [F1], every nonzero proper ideal of R is principal, hence remains principal after localising at any maximal ideal. The ring is Noetherian by [L1], so [L2] makes R a Dedekind domain.

F1L1L2
2.1

Now [L3] applies to the Dedekind domain R and gives that its class group is trivial.

L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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