ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pass

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.
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Opposite diagonals conserve mass under a curved shear

Example

Assume Countable Choice. Map $[- 1, 1]^{2}$ by the area-preserving shear $Φ (x, y) = (x, y + x^{2} / 4)$ . Triangulate the square once along the southwest-to-northeast diagonal and once along the other diagonal; map both meshes by the same $Φ$ . Every source-target overlap has area $1$ . If the source cell values are $2$ and $0$ , each target average is $1$ , source and target total mass are both $4$ , and all target values remain nonnegative.

Facts & Assumptions

Given: Countable Choice; the square, the shear $Φ (x, y) = (x, y + x^{2} / 4)$ , opposite-diagonal triangulations, and source values $2, 0$ .

[F1]

Curved-domain geometry approximation errors: geometry maps carry a Jacobian and define the physical image domain.

[F2]

Intersection-volume conservative remapping: under Countable Choice, target values are exact overlap-weighted averages.

[F3]

Overlap remapping is an L2 projection, conservative and positive: assuming Countable Choice, exact overlap remapping conserves mass and preserves nonnegativity.

[F4]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: under Countable Choice, a $C^{1}$ diffeomorphism changes variables with its positive Jacobian.

[A1]

Countable Choice is used through [F2], [F3], and [F4] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

1.1F1F4A1givenalgebra

The shear is a global $C^{1}$ diffeomorphism with inverse $(X, Y) \mapsto (X, Y - X^{2} / 4)$ and derivative $(\begin{array}{cc} 1 & 0 \\ x / 2 & 1 \end{array})$ of determinant $1$ . Apply [F4] to each overlap to see that its physical area equals the reference overlap area. In the reference square, the two diagonals meet at the center; the two diagonals divide the square into four corner triangles, each with base $2$ along a side and altitude $1$ to the center, so each nonempty source-target overlap has area $1$ .

2.1F2F3step 1.1algebra

Each target triangle has area $2$ and receives source mass $2 (1) + 0 (1) = 2$ , so its target average is $1$ . The source mass is $2 \cdot 2 + 0 \cdot 2 = 4$ , and target mass is $2 (1) + 2 (1) = 4$ .

3.1F2step 2.1∎

The exact remap therefore has source values $(2, 0)$ , four nonempty overlaps of area $1$ , target cell areas $2$ , and target averages $(1, 1)$ . This explicitly verifies the remap formula.

Depends on

Used by

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Sources

Devon Powell and Tom Abel, An exact general remeshing scheme applied to physically conservative voxelization (standard reference, not scraped)
Zhengru Zhang, Moving Mesh Method with Conservative Interpolation Based on L2-Projection, Commun. Comput. Phys. 1 (2006), 930–944 (standard reference, not scraped)