Alphabeta Math
LemmaStatement: 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.

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 FF with V(F)≠∅V(F)\ne\varnothing.

[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 FF has degree greater than one.

assume-contra
2.1

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

step 1.1F1
3.1

Hence FF contains a cycle.

step 2.1L2
4.1

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

step 3.1L1discharge-contradiction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 12 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