ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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 p-adic expansion of minus one

Example

In $Q_{p}$ one has

$- 1 = (p - 1) + (p - 1) p + (p - 1) p^{2} + \dots .$

Facts & Assumptions

Given: A prime $p$ .

[L1]

Every $p$ -adic number has a unique digit expansion (Every p-adic number has a unique digit expansion).

Verification

technique · direct

1.1

The $N$ th partial sum is $(p - 1) (1 + p + \dots + p^{N}) = p^{N + 1} - 1.$ Its difference from $- 1$ is $p^{N + 1}$ , whose $p$ -adic absolute value tends to $0$ . Hence the series converges to $- 1$ .

L1givenalgebra

2.1

Every digit is $p - 1$ , so by the uniqueness part of [L1] this is the digit expansion of $- 1$ .

step 1.1∎

Depends on

Every p-adic number has a unique digit expansion

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, Algebraic Number Theory, Chapter 7 (standard reference, not scraped)