ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 sparse $P_{5}$ -free graph with an anticomplete two-blockade

Example

Let $G$ be the disjoint union of two copies of $P_{3}$ . Then $G$ is $P_{5}$ -free and the two components form an anticomplete two-blockade.

Facts & Assumptions

Given: The graph $G = P_{3} ⊔ P_{3}$ with components $A$ and $B$ .

Verification

technique · direct finite check

1.1

The sets $A$ and $B$ are disjoint induced copies of $P_{3}$ , so they are anticomplete and each has size $3$ . Hence $(A, B)$ is an anticomplete $(2, 3)$ -blockade.

given

2.1

Every induced path in $G$ lies inside a single component, so it has at most three vertices. Thus $G$ is $P_{5}$ -free.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VII. The five-vertex path, Lemma 4.4 pattern (standard reference, not scraped)

A sparse P5P5​-free graph with an anticomplete two-blockade