Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-07-31
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.

A worked graph records vertex deletion, edge deletion, edge contraction and the resulting minor model

Example

Let G have vertices 1,2,3,4 and edges 12,23,34,41,13. Then:

  • G−4 is the triangle on 1,2,3;
  • G−13 is the cycle C4;
  • contracting 12 produces the triangle on the new vertex x and vertices 3,4, so this triangle is a minor of G.

The contraction deletes the loop arising from 12 and merges the two copies of x3 arising from 13 and 23.

1234G123G¡41234G¡13=C4x34G=12:atriangleminor

Facts & Assumptions

Given: The graph G displayed in the Example.

[F1]

Vertex deletion takes the induced subgraph on the remaining vertices, edge deletion removes the named edge, and contraction deletes loops and merges parallel images (Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors, Subgraphs, induced subgraphs and spanning subgraphs).

Verification

technique · direct
1.1

Deleting vertex 4 retains exactly 12,23,13, the three edges of a triangle on 1,2,3. Deleting edge 13 retains exactly 12,23,34,41, the four-cycle.

givenF1
1.2

Under contraction of 12 to x, the edge 12 becomes a loop and is deleted; 13 and 23 both become x3 and merge; 41 becomes 4x; and 34 remains 34. The resulting simple edge set is {x3,34,4x}, a triangle.

givenF1
2.1

Since this triangle is obtained from G by an allowed edge contraction, it is a minor of G.

step 1.2F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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