ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

A closed uncomplemented subspace

Example

Assuming ${A C}_{ω}$ , the inclusion $c_{0} \subseteq ℓ^{\infty}$ is a concrete closed subspace that is not complemented.

Facts & Assumptions

Given: The standard inclusion $c_{0} \subseteq ℓ^{\infty}$ .

[F1]

$c_{0}$ is closed in $ℓ^{\infty}$ (c_0 is a closed subspace of ell-infinity).

[F2]

$c_{0}$ is not complemented in $ℓ^{\infty}$ under ${A C}_{ω}$ (c_0 is not complemented in ell-infinity).

Verification

technique · direct

1.1

The inclusion in the statement is a closed-subspace inclusion by [F1].

F1given

2.1

It is uncomplemented by [F2], so it has both advertised properties.

F2given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Piotr Hajlasz, Functional Analysis, Theorem 10.19 (standard reference, not scraped)