Alphabeta Math
ExampleConstruction: AI-generatedVerification: 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.

In K2,3K_{2,3}, the two vertices in the two-part have three internally disjoint paths and a minimum separator of size three

Example

Let K2,3K_{2,3} have two-part {x1,x2}\{x_1,x_2\} and three-part {y1,y2,y3}\{y_1,y_2,y_3\}. The paths x1yix2x_1y_ix_2 for i=1,2,3i=1,2,3 are internally vertex-disjoint, and {y1,y2,y3}\{y_1,y_2,y_3\} is a minimum x1x_1-x2x_2 separator.

Facts & Assumptions

Given: The displayed complete bipartite graph K2,3K_{2,3}.

[L1]

For nonadjacent terminals, finite vertex Menger equates maximum internally disjoint paths and minimum vertex separators (Menger's theorem: the finite directed and undirected arc, edge and nonadjacent-vertex forms).

Verification

Verification technique: direct.

1.1

The three paths x1y1x2x_1y_1x_2, x1y2x2x_1y_2x_2, and x1y3x2x_1y_3x_2 have distinct internal vertices.

given
1.2

Deleting all three yiy_i destroys every x1x_1-x2x_2 path, while deleting fewer leaves some yiy_i and its two-edge path.

1.3

The terminals are nonadjacent, so [L1] agrees with the direct calculation: both the packing and separator numbers equal three.

L1
2.1

This is a concrete equality case for the local undirected vertex form.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 15 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.