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.

De Rham integration cochain on a smooth path

Example

For a smooth singular path $σ : [0, 1] \to M$ and a smooth one-form $ω$ on $M$ , the de Rham integration cochain evaluates as $I_{M}^{1} (ω) (σ) = \int_{0}^{1} ω_{σ (t)} (σ^{'} (t)) d t .$

Facts & Assumptions

Given: The path and one-form in the example.

[F1]

De Rham integration cochain defines $I_{M}^{1} (ω) (σ) = \int_{σ} ω$ as the integral of the pullback form on the oriented standard one-simplex $[0, 1]$ .

Verification

1.1

At $t \in [0, 1]$ , the pullback definition gives $(σ^{*} ω)_{t} (\partial_{t}) = ω_{σ (t)} (d σ_{t} (\partial_{t})) = ω_{σ (t)} (σ^{'} (t)) .$ Therefore $σ^{*} ω = f (t) d t$ with $f (t) = ω_{σ (t)} (σ^{'} (t))$ .

F1given

2.1

Integrating this coefficient in the positive orientation of $[0, 1]$ and using [F1] gives the displayed formula. If $σ$ is constant then $σ^{'} = 0$ and both sides are zero; if $ω = 0$ the same holds. Both parameter endpoints are included in the smooth-simplex convention and do not change the Riemann integral. The formula uses a single supplied path and no choice principle.

F1step 1.1∎

Depends on

De Rham integration cochain

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

Peter S. Park, Proof of de Rham's Theorem, §3, PDF p.5 (standard reference, not scraped)