ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

$\tau(K_4)=16$ by direct Prüfer enumeration and Cayley's formula

Statement

The complete graph $K_4$ has $16$ spanning trees.

Facts & Assumptions

Given: $K_4$ on the label set $4$ .

[L1]

Its spanning trees correspond bijectively to words of length $2$ on $4$ (Prüfer encoding and decoding are inverse bijections between labelled trees on $n\ge2$ vertices and words of length $n-2$ on their labels).

[L2]

Cayley's formula gives $\tau(K_n)=n^{n-2}$ (Cayley's formula: $\tau(K_n)=n^{n-2}$ for $n\ge2$ , with $\tau(K_1)=1$ and $\tau(K_0)=0$ ).

[F1]

$\tau$ counts spanning trees (The spanning-tree number $\tau(G)$ ).

Verification

technique · direct

1.1

There are $4$ choices for each of the two positions of a Prüfer word, hence $4^2=16$ words and therefore $16$ spanning trees.

L1F1

2.1

Independently, Cayley's formula gives $\tau(K_4)=4^{4-2}=16$ , agreeing with the enumeration.

L2∎

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: 48 results over 22 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.

Sources

ISI Bangalore discrete mathematics notes, Trees and Cayley’s theorem (standard reference, not scraped)

τ(K4)=16\tau(K_4)=16τ(K4​)=16 by direct Prüfer enumeration and Cayley's formula