ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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.

$P_4$ is both connected and anticonnected

Statement

The path $P_4$ is connected and anticonnected. Indeed, $\overline{P_4}\cong P_4$ .

Facts & Assumptions

Given: $P_4$ with vertices $1,2,3,4$ and edges $12,23,34$ .

[F1]

A path is connected (Connected graphs and connected components defined by the existence of vertex paths, Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

[F2]

A graph is anticonnected when its complement is connected (Anticonnected graphs and anticonnected components).

[F3]

Complement edges are precisely the nonedges between distinct vertices (Graph isomorphisms, automorphisms and graph complements).

Verification

technique · direct

1.1

The graph $P_4$ is connected.

1.2

Its complement has edges $13,14,24$ , which form the path $3,1,4,2$ . Thus $\overline{P_4}\cong P_4$ and is connected.

2.1

Therefore $P_4$ is anticonnected as well as connected.

step 1.1step 1.2F2∎

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Cographs survey (Princeton PDF) (standard reference, not scraped)

P4P_4P4​ is both connected and anticonnected