ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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.

An irregular pentagon computed by triangulation and the shoelace formula

Example

Let $P$ have counterclockwise vertices $(0, 0), (4, 0), (5, 2), (2, 4), (0, 3)$ . It is a convex simple pentagon with Jordan content $15$ .

Facts & Assumptions

Given: The displayed cyclic vertex list.

[L1]

A simple polygon has content equal to the sum of the contents of the triangles in any triangulation (A simple polygon is Jordan measurable and its content is the sum of the contents of its triangles).

[L2]

The counterclockwise shoelace formula gives the Jordan content of a simple polygon (The shoelace formula for the area of a counterclockwise simple polygon).

Verification

technique · direct

1.1

Write the vertices as $v_{0}, \dots, v_{4}$ in the displayed order. The five supporting-line calculations $\det (v_{i + 1} - v_{i}, v_{j} - v_{i}) > 0 (j \notin {i, i + 1})$ give respectively the positive lists $(8, 16, 12)$ , $(8, 8, 11)$ , $(16, 8, 7)$ , $(6, 10, 7)$ , and $(12, 15, 6)$ , with indices modulo five. Thus every other vertex lies strictly to the left of every directed boundary edge. The intersection of these five closed left half-planes is convex, has exactly the displayed boundary chain, and contains both diagonals from $v_{0}$ ; hence the chain is simple and counterclockwise and those diagonals triangulate it. The three triangle determinants are $8$ , $16$ , and $6$ , so [L1] gives content $(8 + 16 + 6) / 2 = 15$ .

L1algebra

1.2

The cyclic shoelace sum is $(0 + 8 + 20 + 6 + 0) - (0 + 0 + 4 + 0 + 0) = 30$ , so [L2] gives content $30 / 2 = 15$ .

L2algebra

2.1

The triangulation and shoelace computations agree at $15$ .

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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