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.

A labelled tree and its Prüfer code computed in both directions

Statement

On labels $5 = {0, 1, 2, 3, 4}$ , the tree with edges

${02, 12, 23, 24}$

has Prüfer code $(2, 2, 2)$ , and decoding this word returns the same tree.

Facts & Assumptions

Given: The displayed labelled graph $T$ .

[F1]

Encoding deletes the least leaf and records its neighbour (The Prüfer code of a tree on a finite set of natural labels).

[L1]

The decoding procedure produces the inverse tree (The Prüfer decoding procedure is well defined and produces a labelled tree, Prüfer encoding and decoding are inverse bijections between labelled trees on $n \geq 2$ vertices and words of length $n - 2$ on their labels).

[F2]

At each decoding stage, the least unused label absent from the remaining word is joined to its first symbol; the final two unused labels are then joined.

Verification

technique · direct computation

1.1

Initially the leaves are $0, 1, 3, 4$ . Delete $0$ and record $2$ ; then delete $1$ and record $2$ ; then delete $3$ and record $2$ . Thus the code is $(2, 2, 2)$ .

1.2

To decode, the least label absent from $(2, 2, 2)$ is $0$ , so add $02$ ; for the suffix $(2, 2)$ choose $1$ and add $12$ ; for $(2)$ choose $3$ and add $23$ .

2.1

The unused labels are then $2, 4$ , so add $24$ . The recovered edge set is exactly ${02, 12, 23, 24}$ .

step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

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