ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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 worked graph records vertex deletion, edge deletion, edge contraction and the resulting minor model

Example

Let $G$ have vertices $1,2,3,4$ and edges $12,23,34,41,13$ . Then:

$G-4$ is the triangle on $1,2,3$ ;
$G-13$ is the cycle $C_4$ ;
contracting $12$ produces the triangle on the new vertex $x$ and vertices $3,4$ , so this triangle is a minor of $G$ .

The contraction deletes the loop arising from $12$ and merges the two copies of $x3$ arising from $13$ and $23$ .

Facts & Assumptions

Given: The graph $G$ displayed in the Example.

[F1]

Vertex deletion takes the induced subgraph on the remaining vertices, edge deletion removes the named edge, and contraction deletes loops and merges parallel images (Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors, Subgraphs, induced subgraphs and spanning subgraphs).

Verification

technique · direct

1.1

Deleting vertex $4$ retains exactly $12,23,13$ , the three edges of a triangle on $1,2,3$ . Deleting edge $13$ retains exactly $12,23,34,41$ , the four-cycle.

givenF1

1.2

Under contraction of $12$ to $x$ , the edge $12$ becomes a loop and is deleted; $13$ and $23$ both become $x3$ and merge; $41$ becomes $4x$ ; and $34$ remains $34$ . The resulting simple edge set is $\{x3,34,4x\}$ , a triangle.

givenF1

2.1

Since this triangle is obtained from $G$ by an allowed edge contraction, it is a minor of $G$ .

step 1.2F1∎

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: 4 results over 4 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

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)