LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

Every walk contains a path between the same endpoints, of no greater length

Statement

Every walk contains a path between the same endpoints, of no greater length.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

A simple graph is a pair $(V, E)$ with $V$ any set and $E$ a set of two-element subsets of $V$ (Simple graphs on an arbitrary vertex set).

[F2]

A walk of length $ℓ$ in a simple graph is a finite vertex list $(v_{0}, \dots, v_{ℓ})$ with consecutive vertices adjacent; a path is a walk with distinct vertices; the graph is connected when it is nonempty and every two vertices are joined by a path (Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set).

[L1]

Every nonempty subset $S \subseteq N$ has a least element: there is $ℓ \in S$ with $ℓ \leq s$ for all $s \in S$ . (The well-ordering principle).

Proof

technique · contradiction

1.1

Among the walks from $u$ to $v$ of length at most that of the given one, the well-ordering principle selects one of least length.

F1F2L1choose

2.1

If a vertex occurred twice in the selected walk, deleting the segment between the two occurrences would give a shorter such walk, contradicting minimality.

F2step 1.1assume-contra

3.1

So the selected walk repeats no vertex and is a path with the same endpoints and no greater length; finiteness of the vertex set is never used.

F2step 1.1step 2.1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

15 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)