ExampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

$K_{4}$ decomposed into three complete bipartite graphs, and no decomposition into two

Example

The complete graph $K_{4}$ has the edge decomposition

$K_{{0}, {1, 2, 3}}, K_{{1}, {2, 3}}, K_{{2}, {3}} .$

Facts & Assumptions

Given: the three complete bipartite graphs above.

[L1]

Every complete bipartite decomposition of $K_{n}$ has at least $n - 1$ parts (Graham–Pollak: a complete bipartite decomposition of $K_{n}$ has at least $n - 1$ parts).

Verification

technique · direct

1.1

The three displayed graphs cover the edges ${0, 1}, {0, 2}, {0, 3}$ ; then ${1, 2}, {1, 3}$ ; then ${2, 3}$ , so every edge of $K_{4}$ is covered exactly once.

given

2.1

Hence $K_{4}$ has a complete bipartite decomposition with three parts. Since $n - 1 = 3$ , [L1] says that no decomposition into two parts can exist.

L1step 1.1∎

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

J. Matousek, Thirty-three Miniatures, Miniature 8 (standard reference, not scraped)

K4K4​ decomposed into three complete bipartite graphs, and no decomposition into two