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.

An infinite separated subset of the unit sphere

Example

Assume Dependent Choice. Let $X$ be a normed space that is not the span of any finite list. Then the unit sphere of $X$ contains an infinite subset whose distinct points are all more than $1 / 2$ apart.

Facts & Assumptions

Given: Dependent Choice and a normed space $X$ that is not the span of any finite list.

[L1]

Under these hypotheses, for every $0 < α < 1$ there is a sequence of unit vectors $(x_{n})$ with $∥ x_{n} - x_{m} ∥ > α$ for $n \neq m$ (Under dependent choice, Riesz lemma builds an infinite separated sequence in the unit sphere).

Verification

technique · direct

1.1

Apply [L1] with $α = 1 / 2$ . This gives a sequence $(x_{n})$ of unit vectors such that $∥ x_{n} - x_{m} ∥ > 1 / 2$ whenever $n \neq m$ .

2.1

The set ${x_{n} : n \in N}$ is therefore an infinite subset of the unit sphere whose distinct points are pairwise more than $1 / 2$ apart.

step 1.1∎

Depends on

Under dependent choice, Riesz lemma builds an infinite separated sequence in the unit sphere

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

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)