LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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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.

The path metric on a simplicial tree is geodesic and integer-valued

Statement

For a simplicial tree $T$ , the path metric $d_{T}$ takes values in $N$ , and for every two vertices $v, w$ the unique reduced path from $v$ to $w$ has length exactly $d_{T} (v, w)$ .

Facts & Assumptions

Given: A simplicial tree $T$ .

[L1]

The simplicial path metric is defined to be the length of the unique reduced path joining two vertices. (The simplicial path metric on a tree)

[L2]

Every two vertices of a simplicial tree are joined by a unique reduced path. (A simplicial graph is a tree exactly when every two vertices are joined by a unique reduced path)

Proof

technique · direct

1.1

By [L2], the reduced path from $v$ to $w$ exists and is unique. Its length is an integer, and [L1] defines $d_{T} (v, w)$ to be exactly that integer. So $d_{T}$ is integer-valued.

L1L2given

2.1

Let $v = v_{0}, e_{1}, \dots, e_{n}, v_{n} = w$ be the unique reduced path. Every subpath is again reduced, so [L1] gives $d_{T} (v_{i}, v_{j}) = j - i$ for $i \leq j$ . Hence the path realizes distance on each of its segments, which is the geodesic property claimed in the statement.

L1L2step 1.1algebra∎

Depends on

Used by

Cited to discharge well-definedness by The simplicial path metric on a tree.

Dependency tree · two levels

3 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

Jean-Pierre Serre, Trees (standard reference, not scraped)