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

Kruskal's and Prim's procedures on the same weighted graph

Statement

Let G have edges ab,bc,cd,ad,ac of weights 1,2,3,4,5, respectively. Kruskal's procedure chooses ab,bc,cd. Prim's procedure started at a makes the same choices, and the resulting MST has total weight 6.

abcd45123choseninorder:ab,bc,cd;totalweight6KruskalandPrimfroma

Facts & Assumptions

Given: The weighted graph (G,w) above.

[L1]

Kruskal adds the lightest edge that does not form a cycle (Kruskal's greedy edge procedure produces a minimum spanning tree).

[L2]

Prim adds the lightest edge leaving the reached set (Prim's growing-tree procedure produces a minimum spanning tree).

[F1]

Tree weight is the sum of its edge weights (Real edge-weighted graphs, total tree weight and minimum spanning trees).

Verification

technique · direct computation
1.1L1

Kruskal considers the edges in weight order and accepts ab, then bc, then cd; at that point all four vertices are connected.

1.2L2

From a, Prim first chooses ab of weight 1, then bc of weight 2, then cd of weight 3.

2.1step 1.1step 1.2F1L1L2∎

Both procedures output the path with total weight 1+2+3=6, and their correctness theorems make it an MST.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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