ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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-vertex path has only trivial modules

Example

Write the four-vertex path as $0$ - $1$ - $2$ - $3$ . Then every module of $P_{4}$ is trivial, so $P_{4}$ is prime.

Facts & Assumptions

Given: The path $P_{4}$ with vertices $0, 1, 2, 3$ and edges $01, 12, 23$ .

[L1]

A vertex set is a module when every outside vertex is adjacent to all of it or to none of it (Modules of a graph, and the trivial modules).

[L2]

A graph is prime when its only modules are the trivial ones (Prime graphs: those whose only modules are the trivial ones).

Verification

technique · direct

1.1

Each two-element subset is split by an outside vertex: $2$ splits ${0, 1}$ , $3$ splits ${0, 2}$ , $1$ splits ${0, 3}$ , $3$ splits ${1, 2}$ , $0$ splits ${1, 3}$ , and $1$ splits ${2, 3}$ . So no two-element subset is a module by [L1].

L1given

1.2

Each three-element subset is split by its remaining vertex: $3$ splits ${0, 1, 2}$ , $2$ splits ${0, 1, 3}$ , $1$ splits ${0, 2, 3}$ , and $0$ splits ${1, 2, 3}$ . So no three-element subset is a module.

L1given

2.1

The only modules left are $\emptyset$ , the singletons, and the whole vertex set, so [L2] makes $P_{4}$ prime.

step 1.1step 1.2L2∎

Depends on

Used by

Two disjoint modules whose union is not a module Counterexample
Up to isomorphism the four-vertex path is the only prime graph on four vertices Example
Every graph with at least four vertices has a nontrivial module False statement

Dependency tree · two levels

12 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

M. Habib and C. Paul, A Survey on Algorithmic Aspects of Modular Decomposition, sec. 2.4 (standard reference, not scraped)