Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 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.

A rank-two module and its ideal-class label

Example

Assume the Axiom of Choice. Let R be a Dedekind domain and let I be an invertible fractional ideal. Then the module

M:=R⊕I

is a finite torsion-free module of rank 2, and under the class-group identification its second summand contributes the class [I]∈Cl⁡(R).

Facts & Assumptions

Given: A Dedekind domain R and an invertible fractional ideal I.

[L1]

Every finite torsion-free Dedekind module splits as a finite direct sum of invertible fractional ideals (Finite torsion-free Dedekind modules split into invertible ideal summands).

[L2]

The ideal class group agrees with the Picard group of rank-one projectives (The ideal class group is the Picard group of rank-one projectives).

Verification

technique · direct
1.1

The module M=R⊕I is already displayed as a direct sum of two invertible ideal summands, so it is finite torsion-free and fits the decomposition pattern of [L1].

L1given
2.1

Under the identification of [L2], the free summand R contributes the neutral Picard class and the other summand contributes exactly the class of I. Thus the rank-two module M is labelled by the same ideal class [I].

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