Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

The directed 3-cycle is a strongly connected tournament with a directed Hamilton cycle

Example

Let T have vertices 1,2,3 and arcs

1→2,2→3,3→1.

Then T is a strongly connected tournament, and 1,2,3,1 is a directed Hamilton cycle.

123

Facts & Assumptions

Given: The three explicitly displayed arcs.

[F1]

A tournament has exactly one directed arc between each pair of distinct vertices (A tournament is an orientation of a complete finite graph).

[F2]

Strong connectivity requires a directed path for each ordered vertex pair, and a directed Hamilton cycle visits every vertex once before returning (Directed walks, trails, paths and cycles, and strong connectivity).

[L1]

Camion's theorem gives a directed Hamilton cycle in every strong tournament of order at least three (Camion's theorem: every strongly connected tournament on at least three vertices has a directed Hamilton cycle).

Verification

technique · direct
1.1

The three arcs orient the three unordered vertex pairs exactly once, so T is a tournament. The ordered pairs (1,2),(2,3),(3,1) are joined directly, while (2,1),(3,2),(1,3) are joined by the two-arc paths through the remaining vertex. Hence T is strongly connected.

givenF1F2
1.2

The closed list 1,2,3,1 follows the three arcs and visits every vertex once before returning, so it is a directed Hamilton cycle, as [L1] predicts.

givenF2L1
2.1

This verifies all three assertions in the example.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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