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 same sequence behaves oppositely in the real and p-adic metrics

Example

For a fixed prime $p$ , the sequence $(p^{n})_{n \geq 0}$ tends to $\infty$ in the usual absolute value and to $0$ in $Q_{p}$ .

Facts & Assumptions

Given: A prime $p$ .

[L1]

$∣ p ∣_{p} = p^{- 1}$ by definition of the $p$ -adic absolute value (The p-adic absolute value on the rationals).

[L2]

$Q_{p}$ is the completion field for this metric (The p-adic numbers as a metric completion).

Verification

technique · direct

1.1

By [L1], $∣ p^{n} ∣_{p} = p^{- n} \to 0$ , so the sequence converges to $0$ in the $p$ -adic metric and hence in the completion $Q_{p}$ .

L1L2givenalgebra

2.1

In the usual absolute value, $∣ p^{n} ∣_{\infty} = p^{n} \to \infty$ , so the same sequence runs away instead of converging to $0$ .

step 1.1algebra∎

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

Andrew V. Sutherland, 18.782 Lecture 5 (standard reference, not scraped)