ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 small graph separates walks, trails, paths, closed walks and cycles by explicit vertex lists

Example

Let $G$ have vertex set ${0, 1, 2, 3}$ and edge set ${01, 12, 20, 23}$ , where $i j$ abbreviates ${i, j}$ . The following lists separate the standard notions:

$(0, 1, 2, 1)$ is a walk but not a trail;
$(3, 2, 0, 1, 2)$ is a trail but not a path;
$(3, 2, 1, 0)$ is a path;
$(0, 1, 0)$ is a closed walk but not a cycle;
$(0, 1, 2, 0)$ is a cycle.

Facts & Assumptions

Given: The graph $G$ and the vertex lists displayed above.

[F1]

Walks require adjacent successive vertices; trails do not repeat edges; paths do not repeat vertices; cycles repeat only their first vertex at the end and have length at least $3$ (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

[F2]

The displayed edge set is a subset of the two-element subsets of the finite vertex set, so $G$ is a finite simple graph (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets).

Verification

technique · direct

1.1

Every successive pair in every displayed list is one of $01, 12, 20, 23$ , so every list is a walk in $G$ .

givenF1F2

2.1

The list $(0, 1, 2, 1)$ traverses $12$ twice, so it is not a trail. The list $(3, 2, 0, 1, 2)$ traverses the distinct edges $23, 20, 01, 12$ but repeats vertex $2$ , so it is a trail and not a path.

step 1.1F1

2.2

The list $(3, 2, 1, 0)$ has four distinct vertices, so it is a path. The list $(0, 1, 0)$ is closed but repeats the edge $01$ and has length $2$ , so it is not a cycle.

step 1.1F1

3.1

The list $(0, 1, 2, 0)$ is closed, has length $3$ , and has distinct vertices before returning to $0$ , so it is a cycle.

step 1.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

University of Lethbridge, Combinatorics: A Guided Tour, Paths and Cycles (standard reference, not scraped)