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.

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 $\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 8 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.