Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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.

Edge-disjoint closed trails sharing a vertex can be spliced into one closed trail

Statement

Two edge-disjoint closed trails in an undirected multigraph that share a vertex can be spliced into a closed trail using exactly the edges of both. The same holds for two arc-disjoint directed closed trails in a digraph.

Facts & Assumptions

Given: Closed trails CC and DD sharing a vertex ww, with disjoint edge sets or disjoint arc sets as appropriate.

[F1]

A trail repeats no edge or arc, and a closed trail has the same initial and terminal vertex (Euler trails and Euler circuits in multigraphs and digraphs).

[F2]

A directed walk respects the orientation of every traversed arc (Directed walks, trails, paths and cycles, and strong connectivity).

Proof

technique · constructive
1.1

Rotate the cyclic listing of each closed trail so that it starts and ends at ww. Traverse CC from ww back to ww, then traverse DD from ww back to ww. Consecutive edges or arcs still meet at their displayed vertices, and directed arcs retain their orientations.

givenF1F2construct
2.1

The concatenation is closed, and it repeats no edge or arc because neither input trail repeats one and their used sets are disjoint. Its used set is exactly the union of the two input used sets.

step 1.1F1given
3.1

This concatenation is the required spliced closed trail in either setting.

step 1.1step 2.1discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 8 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