ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

$k^{- a}$ membership in $ℓ^{p}$

Example

For $a > 0$ , define a sequence on $N$ by $a_{0} : = 0$ and $a_{k} : = k^{- a}$ for $k \geq 1$ . Then $(a_{k})_{k \geq 0}$ belongs to $ℓ^{p}$ exactly when

$a p > 1.$

Facts & Assumptions

Given: Real numbers $a > 0$ and $p > 0$ .

[L1]

$ℓ^{p}$ is the counting-measure version of $L^{p}$ ( $ℓ^{p}$ is the $L^{p}$ space of counting measure).

[L2]

The real $p$ -series $\sum_{k \geq 1} k^{- s}$ converges exactly when $s > 1$ (The p-series for a real exponent p converges exactly when p is greater than one).

Verification

Proof technique: Apply the real $p$ -series test to the series $\sum k^{- a p}$ and read off the threshold $a p > 1$ .

1.1

By [L1], the sequence $(a_{k})$ lies in $ℓ^{p}$ exactly when [L1, given] $\sum_{k = 0}^{\infty} ∣ a_{k} ∣^{p} = \sum_{k = 1}^{\infty} k^{- a p}$ converges.

2.1

By [L2], that series converges exactly when $a p > 1$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorems 8.12 and 8.13 (standard reference, not scraped)
John K. Hunter, Measure Theory, Chapter 17 (standard reference, not scraped)

k−ak−a membership in ℓpℓp