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.

Sections on an open subset extended by the empty set outside it

Example

Fix an open subset $W \subseteq X$ . Define a presheaf $E_{W}$ on $X$ by $E_{W} (V) = {\begin{cases} {*}, & V \subseteq W, \\ \emptyset, & V \subseteq̸ W . \end{cases}$ Then $E_{W}$ is a sheaf of sets on $X$ . It is the simplest example of data carried on an open subset and extended by no sections outside that subset.

Facts & Assumptions

Given: An open subset $W \subseteq X$ and an open cover $V = ⋃_{i \in I} V_{i}$ .

[L1]

A sheaf is a presheaf with locality and unique gluing on every open cover (A sheaf on a topological space).

Verification

technique · direct

1.1

If $V \subseteq W$ , then every $V_{i} \subseteq W$ has the unique section $*$ , and the only possible glued section on $V$ is again $*$ . If $V \subseteq̸ W$ , choose $x \in V ∖ W$ . Some cover member $V_{i}$ contains $x$ , so $V_{i} \subseteq̸ W$ and $E_{W} (V_{i}) = \emptyset$ . Hence there is no compatible family of local sections on this cover. In either case the gluing and uniqueness clauses in [L1] are satisfied.

L1given

2.1

Restriction maps are forced: from ${*}$ to ${*}$ they are the identity, and into $\emptyset$ there is only the empty function from $\emptyset$ . Thus $E_{W}$ is a presheaf, and step 1.1 shows it is a sheaf.

step 1.1construct∎

Depends on

A sheaf on a topological space

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

The Stacks Project, Sheaves on Spaces, Section 31 (standard reference, not scraped)