Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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 plane forest has exactly one face

Statement

Every polygonally embedded finite forest has exactly one face. Forests and leaves are those of Trees, forests, leaves and isolated vertices, and the finite edge-component identity is For every forest, ∣V∣=∣E∣+c, where c is the number of connected components; Every nonempty forest has a vertex of degree at most one supplies the deletion step.

Facts & Assumptions

Given: A plane forest F.

[L1]

For a finite forest, ∣V(F)∣=∣E(F)∣+c(F) (For every forest, ∣V∣=∣E∣+c, where c is the number of connected components).

[L2]

A bridge borders one face, counted on both local sides (Face frontiers are unions of whole edges; a cycle edge borders two faces and a bridge borders one).

Proof

technique · induction
1.1

The null forest and a forest of isolated vertices have connected complement and one face. In a nonempty forest with an edge, a low-degree vertex lemma supplies a leaf and its incident edge.

baseL1
1.2

Delete a leaf and its edge. The remaining drawing is a smaller plane forest. By the induction hypothesis it has one face, and reinserting the pendant edge does not split that face because the edge is a bridge and both its local sides are incident with the same face by [L2].

ihL2
2.1

Repeating the leaf deletion reaches isolated vertices, so every plane forest has one face.

step 1.1step 1.2discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 21 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