ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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

$α = (0 1 2) (3 4), β = (0 3) (5 6), σ = α β .$

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

$σ = (0 4 3 1 2) (5 6) .$

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 $β$ and then $α$ gives $σ (0) = 4$ , $σ (1) = 2$ , $σ (2) = 0$ , $σ (3) = 1$ , $σ (4) = 3$ , $σ (5) = 6$ , and $σ (6) = 5$ , so the one-line form is $[4, 2, 0, 1, 3, 6, 5]$ .

givenL1

2.1

Starting at $0$ gives $0 \mapsto 4 \mapsto 3 \mapsto 1 \mapsto 2 \mapsto 0$ , and the only unused points satisfy $5 \mapsto 6 \mapsto 5$ . These cycles are disjoint and reproduce every value in step 1.1, so $σ = (0 4 3 1 2) (5 6)$ .

step 1.1L1∎

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.