ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

De rham cohomology of euclidean space

Example

For every $n \geq 0$ , $H_{d R}^{∙} (R^{n}) = R$ in degree zero only.

Facts & Assumptions

Given: Euclidean space with centre $0$ .

[F1]

Poincare lemma for differential forms on star shaped domains: Every closed smooth $k$ -form on a star-shaped open domain is exact for $k \geq 1$ . For centre $0$ , one primitive is $η_{x} (v_{1}, \dots, v_{k - 1}) = \int_{0}^{1} t^{k - 1} ω_{t x} (x, v_{1}, \dots, v_{k - 1}) d t$ .

[F2]

Zero th de rham cohomology is locally constant functions: $H_{d R}^{0} (M)$ is the algebra of locally constant real functions. For nonempty connected $M$ it is canonically $R$ .

Verification

technique · direct

1.1

For any closed $k$ -form with $k \geq 1$ , the radial primitive is $η_{x} (v_{1}, \dots, v_{k - 1}) = \int_{0}^{1} t^{k - 1} ω_{t x} (x, v_{1}, \dots, v_{k - 1}) d t$ , and $d η = ω$ . Thus every positive-degree class is zero.

F1given

2.1

Euclidean space is nonempty and connected, so its degree-zero classes are constant functions with their ordinary multiplication. Negative degrees are zero by the complex convention. When $n = 0$ , the only form space is the constants on a point, giving the same ring.

F2step 1.1∎

Source locator

Lee, Theorem 17.14, p.447, Poincaré lemma, and Proposition 17.6, p.443, degree zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (2014) (standard reference, not scraped)