ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Chain Stokes on an oriented two-simplex

Example

For an oriented smooth two-simplex $σ = [v_{0} v_{1} v_{2}]$ in a smooth manifold and a smooth one-form $η$ , $\int_{σ} d η = \int_{[v_{1} v_{2}]} η - \int_{[v_{0} v_{2}]} η + \int_{[v_{0} v_{1}]} η .$ For the standard triangle in $R^{2}$ and $η = x d y$ , both sides equal $1 / 2$ .

Facts & Assumptions

Given: The oriented simplex and one-form in the example.

[F1]

Stokes theorem for smooth singular chains gives $\int_{c} d η = \int_{\partial c} η$ with the alternating face differential.

Verification

1.1

The oriented boundary is $\partial [v_{0} v_{1} v_{2}] = [v_{1} v_{2}] - [v_{0} v_{2}] + [v_{0} v_{1}] .$ Applying [F1] and linearity of chain integration gives exactly the first displayed formula, including its middle minus sign.

F1given

2.1

For the standard triangle with $v_{0} = (0, 0)$ , $v_{1} = (1, 0)$ and $v_{2} = (0, 1)$ , one has $d (x d y) = d x \land d y$ and therefore $\int_{σ} d η = \int_{0}^{1} \int_{0}^{1 - x} d y d x = \int_{0}^{1} (1 - x) d x = \frac{1}{2} .$ On the first boundary edge use $(x, y) = (1 - t, t)$ , $0 \leq t \leq 1$ , so $x d y = (1 - t) d t$ and its integral is $1 / 2$ . On $[v_{0} v_{2}]$ one has $x = 0$ , and on $[v_{0} v_{1}]$ one has $d y = 0$ , so the other two edge integrals vanish. Thus the signed boundary total is also $1 / 2$ .

step 1.1algebra

3.1

Reversing the simplex orientation reverses both sides and all three induced edge signs. A zero form or degenerate simplex gives zero through [F1]; there is no omitted boundary endpoint because each oriented edge includes both of its vertices. The calculation is finite and choice-free.

F1step 1.1step 2.1∎

Depends on

Stokes theorem for smooth singular chains

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

Peter S. Park, Proof of de Rham's Theorem, Theorem 3.1, PDF p.5 (standard reference, not scraped)