ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Changing a section by a one-cochain

Example

In the direct-product extension

$0 \to Z \to Z \times C_{2} \to C_{2} \to 0,$

the standard section has zero factor set, while changing the nontrivial value by the one-cochain $u (t) = 1$ produces a new factor set with $f^{'} (t, t) = 2$ .

Facts & Assumptions

Given: The direct-product extension $Z \times C_{2}$ with trivial action.

[F1]

The factor set of a section is defined by the equation $s (g) s (h) s (g h)^{- 1} = i (f_{s} (g, h))$ (Normalized set-theoretic section and factor set).

[L1]

Changing the section changes the factor set by a coboundary (Changing the section changes the factor set by a coboundary).

Verification

technique · direct

1.1

For the standard section $s (t) = (0, t)$ , the equality in [F1] gives $f_{s} = 0$ because $(0, t)^{2} = (0, 1)$ .

F1givenalgebra

2.1

Replace the section by $s^{'} (t) = (1, t)$ . Then $(1, t)^{2} = (2, 1)$ , so [F1] gives $f_{s^{'}} (t, t) = 2$ and $f_{s^{'}} (1, t) = f_{s^{'}} (t, 1) = 0$ . Thus $f_{s^{'}} = δ u$ for the one-cochain with $u (t) = 1$ .

F1L1step 1.1algebra

3.1

This makes the section-change formula visible in a concrete computation.

step 2.1∎

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

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)