ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Change of variables on an oriented circle

Example

Fix $a \in R$ with $∣ a ∣ < 1$ . The map $F (e^{i t}) = e^{i (t + a \sin t)}$ is an orientation-preserving smooth circle diffeomorphism. For the standard angular form $ω$ on the unit circle, $F^{*} ω = (1 + a \cos t) d t, \int_{S^{1}} F^{*} ω = 2 π = \int_{S^{1}} ω,$ where t denotes the angle on each cut chart.

Facts & Assumptions

[F1]

Change of variables on oriented manifolds: Let $F : M \to N$ be a diffeomorphism of oriented smooth $n$ -manifolds and $ω \in Ω_{c}^{n} (N)$ . If $F$ preserves orientation everywhere, $\int_{M} F^{*} ω = \int_{N} ω$ ; if it reverses orientation everywhere, $\int_{M} F^{*} ω = - \int_{N} ω$ . If the sign varies between components, apply the appropriate signed equality on each component and add.

[F2]

Computing form integrals by finite parametrizations: Let $n \geq 1$ , let $M^{n}$ be oriented, and let $ω \in Ω_{c}^{n} (M)$ . For $1 \leq i \leq m$ let $D_{i} \subset R^{n}$ be bounded open Jordan domains and $F_{i} : {\overline{D}}_{i} \to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_{i} ∣_{D_{i}}$ is an orientation-preserving diffeomorphism onto an open $W_{i} \subset M$ , the $W_{i}$ are pairwise disjoint, and $supp ω \subset ⋃_{i} {\overline{W}}_{i}$ . Then $\int_{M} ω = \sum_{i = 1}^{m} \int_{D_{i}} F_{i}^{*} ω .$ An empty family is allowed when the support is empty. No nonsingularity of $D F_{i}$ on $\partial D_{i}$ , and no $M$ -valued extension across a genuine target boundary, is assumed.

Verification

Given: The objects and hypotheses in the statement above.

1.1

The lift $h (t) = t + a \sin t$ satisfies $h (t + 2 π) = h (t) + 2 π$ and $h^{'} (t) = 1 + a \cos t \geq 1 - ∣ a ∣ > 0$ . It is strictly increasing; since $∣ h (t) - t ∣ \leq ∣ a ∣$ , its limits are the two infinities, so it is onto R. Its inverse is smooth by the one-dimensional inverse theorem, and both lift maps commute with shifts by $2 π$ . They descend to inverse smooth circle maps, preserving increasing-angle orientation.

algebra

2.1

Pulling back the angular form in cut charts gives $d h = (1 + a \cos t) d t$ . The cut-circle parametrization yields $\int_{0}^{2 π} (1 + a \cos t) d t = 2 π$ , while $\int_{0}^{2 π} d t = 2 π$ . Both cut parametrizations extend smoothly from the closed interval.

F2step 1.1

3.1

The circle is compact, so the form is compactly supported; oriented change of variables therefore predicts the same equality and all its hypotheses have just been checked. For a=0 the map is the identity. Values $∣ a ∣ = 1$ are excluded because the derivative can vanish, so the proof does not assert a diffeomorphism at those endpoints.

F1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Lee Proposition 16.6(d), pp.407–408 (explicit circle specialization) (standard reference, not scraped)