ExampleConstruction: AI-adaptedVerification: 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.

The four-cycle is a cograph

Example

The four-cycle $C_{4}$ is a cograph.

Facts & Assumptions

Given: The cycle $C_{4}$ on vertices $0, 1, 2, 3$ .

[L1]

Any graph built from one-vertex graphs by repeated disjoint unions and complete connections is a cograph (Cographs by the singleton, disjoint-union, and complete-connection recursion).

[L2]

In $C_{4}$ , the edges are exactly $01, 12, 23, 30$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L3]

The complete connection of two vertex-disjoint graphs is obtained by keeping all internal edges and adding every possible cross edge (The complete connection of two disjoint graphs).

Verification

technique · direct construction

1.1

Let $A : = {0, 2}$ and $B : = {1, 3}$ . By [L2], there is no edge inside $A$ and no edge inside $B$ , while every cross pair $0$ - $1$ , $1$ - $2$ , $2$ - $3$ , and $3$ - $0$ is an edge. Thus the induced graphs on $A$ and $B$ are two edgeless two-vertex graphs.

2.1

Each edgeless two-vertex graph is a cograph, because it is the disjoint union of two one-vertex graphs. Step 1.1 and [L3] show that $C_{4}$ is the complete connection of those two cographs. Therefore [L1] makes $C_{4}$ a cograph.

step 1.1L1L3

3.1

Hence the four-cycle is a cograph.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Tero Harju, Lecture Notes on Combinatorial Structures in Graph Theory, Exercise 5.2 (standard reference, not scraped)