LemmaStatement: AI-adaptedProof: AI-generatedprecheck 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.

Every nonempty forest has a vertex of degree at most one

Statement

Every forest with at least one vertex has a vertex of degree at most one.

Facts & Assumptions

Given: A forest $F$ with $V (F) \neq \emptyset$ .

[L1]

A forest contains no cycle (Trees, forests, leaves and isolated vertices).

[L2]

Every finite graph of minimum degree at least two contains a cycle (Every finite simple graph of minimum degree at least two contains a cycle).

[F1]

Vertex degrees are natural numbers (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

Proof

technique · contradiction

1.1

Suppose every vertex of $F$ has degree greater than one.

assume-contra

2.1

Since degrees are natural numbers, $F$ has minimum degree at least two.

step 1.1F1

3.1

Hence $F$ contains a cycle.

step 2.1L2

4.1

This contradicts that $F$ is a forest, so some vertex has degree at most one.

step 3.1L1discharge-contradiction∎

Depends on

Used by

Every plane forest has exactly one face Proposition
For every forest, |V|=|E|+c, where c is the number of connected components Theorem

Dependency tree · two levels

8 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

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)