ExampleConstruction: Literature-sourcedVerification: 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.
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.

Collinear and duplicate sites return a canonical chain

Statement

For unweighted sites with distinct ids at $(0, 0), (1, 0), (1, 0), (2, 0)$ , duplicate collapse retains the least id at $(1, 0)$ , and the affine-dimension-one output is the ordered two-edge chain from $(0, 0)$ to $(2, 0)$ .

Facts & Assumptions

Given: The four listed exact dyadic coordinates, stable identifiers $1, 2, 3, 4$ in that order, and unweighted duplicate handling.

[F1]

Finite indexed geometric site sets and duplicate handling: unweighted duplicates are equal coordinates and each class retains its least identifier.

[F2]

Planar triangulations, Delaunay structures, and Voronoi cells: an affine-dimension-one set is sorted by the first varying coordinate and joined consecutively, with no triangles.

[F3]

Exact planar insertion terminates with a weak empty-circle triangulation: the lower-dimensional branch returns the defined chain.

[F4]

A planar triangulation has linearly many triangles and edges: the planar triangle-count theorem assumes affine dimension two and makes no claim on this branch.

[A1]

The coordinate list is exactly the stated finite input, and its first varying coordinate is $x$ .

Proof

1.1givenF1A1algebra

The two entries at $(1, 0)$ form one unweighted duplicate class by [F1]. Their identifiers are $2$ and $3$ , so the retained representative has id $2$ and id $3$ is recorded as discarded. The deduplicated coordinates are $(0, 0), (1, 0), (2, 0)$ .

2.1step 1.1F2algebra

These three distinct points lie on the horizontal line $y = 0$ and are not all equal, so their affine dimension is exactly one. Sorting by $x$ gives the displayed order, and [F2] joins consecutive points, yielding edges $(0, 0) (1, 0)$ and $(1, 0) (2, 0)$ .

3.1F2F3F4step 2.1∎

The output has no triangles because the affine dimension is one; the algorithm therefore uses the branch of [F3], while the affine-dimension-two premise of [F4] does not apply.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Mark de Berg et al., Computational Geometry: Algorithms and Applications, 3rd ed., Chapter 9, Section 9.1 (standard reference, not scraped)
David M. Mount, CMSC 754 Computational Geometry, Lecture 11 (standard reference, not scraped)