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 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 $ij$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 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

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