ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

From one-line notation to a disjoint-cycle decomposition, with the right-hand factor acting first

Example

In $S_7$ , let

$\alpha=(0\,1\,2)(3\,4),\qquad \beta=(0\,3)(5\,6),\qquad \sigma=\alpha\beta.$

The right-hand factor acts first. The one-line form of $\sigma$ is $[4,2,0,1,3,6,5]$ , and its disjoint-cycle decomposition is

$\sigma=(0\,4\,3\,1\,2)(5\,6).$

Facts & Assumptions

Given: The displayed permutations in $S_7$ , with composition from right to left.

[L1]

A permutation of a finite set is recovered by following each unused point until its orbit closes, producing its disjoint-cycle decomposition (Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation).

Verification

technique · direct

1.1

Applying $\beta$ and then $\alpha$ gives $\sigma(0)=4$ , $\sigma(1)=2$ , $\sigma(2)=0$ , $\sigma(3)=1$ , $\sigma(4)=3$ , $\sigma(5)=6$ , and $\sigma(6)=5$ , so the one-line form is $[4,2,0,1,3,6,5]$ .

givenL1

2.1

Starting at $0$ gives $0\mapsto4\mapsto3\mapsto1\mapsto2\mapsto0$ , and the only unused points satisfy $5\mapsto6\mapsto5$ . These cycles are disjoint and reproduce every value in step 1.1, so $\sigma=(0\,4\,3\,1\,2)(5\,6)$ .

step 1.1L1∎

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: 30 results over 11 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.