ExampleConstruction: AI-generatedVerification: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Primes inside a localization at a prime

Example

Let $A = k [x, y]$ and let $m = (x, y)$ . The prime ideals $(0)$ , $(x)$ , $(y)$ , and $(x + y)$ all lie inside $m$ , so they give distinct prime ideals of the localization $A_{m}$ .

Facts & Assumptions

Given: A field $k$ , the polynomial ring $A = k [x, y]$ , and the prime ideal $m = (x, y)$ .

[L1]

Prime ideals of $A_{m}$ correspond exactly to prime ideals of $A$ contained in $m$ (Primes of a localization at a prime).

Verification

technique · direct

1.1

The ideals $(0)$ , $(x)$ , $(y)$ , and $(x + y)$ are prime in $A$ , and each is contained in $m$ because their generators lie in $(x, y)$ .

givenalgebra

2.1

By [L1], each of these four primes determines a distinct prime ideal of $A_{m}$ , and every prime of $A_{m}$ comes from some prime of $A$ contained in $m$ .

L1step 1.1

3.1

This records a concrete finite family of primes inside the prime localization.

step 2.1∎

Depends on

Primes of a localization at a prime

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)