CorollaryStatement: AI-adaptedProof: 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 tree on $n\ge1$ vertices has $n-1$ edges

Statement

A tree with $n\ge1$ vertices has exactly $n-1$ edges.

Facts & Assumptions

Given: A tree $T$ with $|V(T)|=n$ .

[L1]

Every forest satisfies $|V|=|E|+c$ (For every forest, $|V|=|E|+c$ , where $c$ is the number of connected components).

[F1]

A tree is connected and acyclic, so it is a forest with exactly one connected component (Trees, forests, leaves and isolated vertices, Connected graphs and connected components defined by the existence of vertex paths).

Proof

technique · direct

1.1

Apply the forest identity to $T$ : $n=|E(T)|+1$ .

L1F1

2.1

Therefore $|E(T)|=n-1$ .

step 1.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 19 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)
ISI Bangalore discrete mathematics notes, Trees and Cayley’s theorem (standard reference, not scraped)

A tree on n≥1n\ge1n≥1 vertices has n−1n-1n−1 edges