ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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 fundamental cycle and a fundamental cut in a fixed spanning tree

Statement

Let $G$ have vertices $\{1,2,3,4\}$ and edges $\{12,23,34,14,13\}$ . For the spanning tree $T$ with edges $\{12,23,34\}$ , the edge $14$ has fundamental cycle $1,2,3,4,1$ . The edge $23$ has fundamental cut $\{23,13,14\}$ .

Facts & Assumptions

Given: The graph $G$ and spanning subgraph $T$ above.

[L1]

An outside edge and its unique tree path form the fundamental cycle (Every edge outside a spanning tree determines a unique fundamental cycle).

[L2]

Deleting a tree edge partitions the tree into two sides, whose crossing edges form the fundamental cut (Every edge of a spanning tree determines a fundamental cut, and every edge crossing it restores a spanning tree).

[F1]

A connected acyclic spanning subgraph is a spanning tree (Spanning trees of a graph).

Verification

technique · direct computation

1.1

The graph $T$ is the path $1,2,3,4$ , so it is a spanning tree. Its unique $1$ - $4$ path together with $14$ is the stated $4$ -cycle.

L1F1

1.2

Deleting $23$ leaves vertex sides $\{1,2\}$ and $\{3,4\}$ . Exactly $23,13,14$ cross between them in $G$ .

2.1

Hence the displayed cycle and cut are the required fundamental objects.

step 1.1step 1.2∎

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: 15 results over 9 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

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)