Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-adaptedSession-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.

The spanning-tree exchange lemma

Statement

Let TT be a spanning tree of GG.

  1. If e∈E(G)∖E(T)e\in E(G)\setminus E(T) and ff is any edge other than ee on the fundamental cycle of ee, then T−f+eT-f+e is a spanning tree.
  2. If f∈E(T)f\in E(T) and ee crosses the fundamental cut of ff, then T−f+eT-f+e is a spanning tree.

Facts & Assumptions

Given: A graph GG and a spanning tree TT.

[L1]

T+eT+e has one fundamental cycle for every e∉Te\notin T (Every edge outside a spanning tree determines a unique fundamental cycle).

[L2]

Removing f∈Tf\in T gives a fundamental cut, and adding any crossing edge restores a spanning tree (Every edge of a spanning tree determines a fundamental cut, and every edge crossing it restores a spanning tree).

[F1]

A spanning tree is a connected acyclic spanning subgraph (Spanning trees of a graph).

Proof

technique · constructive
1.1

Construct T′:=T−f+eT':=T-f+e, where e∉E(T)e\notin E(T) and f≠ef\ne e lies on the fundamental cycle of ee. Removing ff breaks that cycle but leaves the graph connected, because the rest of the cycle replaces every use of ff.

constructL1
2.1

No other cycle exists in T+eT+e, so T−f+eT-f+e is connected, acyclic and spanning, hence a spanning tree.

step 1.1L1F1
3.1

The second assertion is exactly the restoration clause for the fundamental cut of ff.

L2discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 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