ExampleConstruction: Literature-sourcedVerification: 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.

The distinguished subset D(x) matches the primes of the localization at x

Example

In the polynomial ring $k [x, y]$ , the distinguished subset $D (x)$ is exactly the set of prime ideals that correspond to prime ideals of the principal localization $k [x, y]_{x}$ .

Facts & Assumptions

Given: A field $k$ and the polynomial ring $k [x, y]$ .

[L1]

$D (x)$ is the set of prime ideals that do not contain $x$ (Principal distinguished subsets of the prime spectrum).

[L2]

Prime ideals of $k [x, y]_{x}$ correspond exactly to prime ideals of $k [x, y]$ that do not contain $x$ (Primes of a principal localization).

Verification

technique · direct

1.1

By [L1], a prime ideal of $k [x, y]$ lies in $D (x)$ exactly when it avoids $x$ .

2.1

By [L2], the same condition characterizes the prime ideals that survive in the localization $k [x, y]_{x}$ . Therefore the contraction map from $Spec (k [x, y]_{x})$ identifies its image with $D (x)$ .

L2step 1.1

3.1

So $D (x)$ is the prime-set avatar of localizing at $x$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 and §17 (standard reference, not scraped)