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.

The path-antipath theorem specialized to $P_{5}$ and $\overline{P_{5}}$

Example

Graphs with no induced $P_{5}$ and no induced $\overline{P_{5}}$ have the Erdős–Hajnal property.

Facts & Assumptions

Given: The standard five-vertex path $P_{5}$ .

[L1]

For every integer $k \geq 2$ , the class forbidding $P_{k}$ and $\overline{P_{k}}$ has the Erdős–Hajnal property (For every $k$ , the class forbidding $P_{k}$ and $\overline{P_{k}}$ has the Erdős–Hajnal property).

[L2]

The graph $P_{5}$ is the standard path on five vertices (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Verification

technique · direct

1.1

By [L2], the parameter choice $k = 5$ in [L1] is exactly the class forbidding $P_{5}$ and $\overline{P_{5}}$ .

L1L2

2.1

Therefore graphs with no induced $P_{5}$ and no induced $\overline{P_{5}}$ have the Erdős–Hajnal property.

step 1.1∎

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.

The path-antipath theorem specialized to P5P5​ and P5‾P5​​