ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

A co-leaf of $\overline{P_{5}}$ is exactly a leaf of $P_{5}$

Example

Let $P_{5}$ have vertices $1, 2, 3, 4, 5$ in path order. In $\overline{P_{5}}$ the vertices $1$ and $5$ are co-leaves, and no other vertex is.

Facts & Assumptions

Given: The five-vertex path $P_{5}$ with edges $12, 23, 34, 45$ .

[L1]

A co-leaf of a graph is a vertex that is a leaf in the complement, equivalently a vertex of degree $∣ V (G) ∣ - 2$ (Co-leaves of a finite graph).

[L2]

The complement contains exactly the nonedges of the original graph (Graph isomorphisms, automorphisms and graph complements).

Verification

technique · direct

1.1

By [L2], the complement $\overline{P_{5}}$ has edges $13, 14, 15, 24, 25, 35$ . So vertices $1$ and $5$ each have degree $3 = 5 - 2$ in $\overline{P_{5}}$ , while vertices $2, 3, 4$ each have degree $2$ .

givenL2algebra

2.1

Step 1.1 and [L1] show that $1$ and $5$ are co-leaves of $\overline{P_{5}}$ , and that $2, 3, 4$ are not. Since $1$ and $5$ are exactly the leaves of the path $P_{5}$ , the co-leaves of $\overline{P_{5}}$ are precisely the leaves of $P_{5}$ .

step 1.1L1∎

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.

A co-leaf of P5‾P5​​ is exactly a leaf of P5P5​