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.

A five-cycle is even and a six-cycle is odd

Example

The cycle $(0 1 2 3 4)$ is even, while $(0 1 2 3 4 5)$ is odd.

Facts & Assumptions

Given: The displayed cycles in symmetric groups on sets containing their entries.

[L1]

A cycle of length $k$ has sign $(- 1)^{k - 1}$ (A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles).

Verification

technique · direct

1.1

The five-cycle is a product of $5 - 1 = 4$ transpositions and has sign $(- 1)^{4} = + 1$ ; the six-cycle is a product of $6 - 1 = 5$ transpositions and has sign $(- 1)^{5} = - 1$ .

givenL1

2.1

Thus the five-cycle is even and the six-cycle is odd. The shift by one comes from the number of transpositions in the standard cycle factorisation.

step 1.1L1∎

Depends on

A $k$-cycle has sign $(-1)^{k-1}$, and $\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}$ when fixed points are counted as cycles

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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.