ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Foata's transformation on the permutation $2 0 3 1$

Example

Foata's transformation sends the permutation $[2, 0, 3, 1]$ to $[2, 3, 0, 1]$ .

Facts & Assumptions

Given: Foata's recursive transformation and the major-index/inversion identity (Foata's recursive transformation on permutations, Foata's transformation sends major index to inversion number).

Verification

technique · constructive

1.1

Applying the recursion step by step gives $Φ ([2]) = [2]$ , then $Φ ([2, 0]) = [2, 0]$ , then $Φ ([2, 0, 3]) = [2, 0, 3]$ , and finally $Φ ([2, 0, 3, 1]) = [2, 3, 0, 1]$ .

construct

2.1

The original permutation has descents at positions $0$ and $2$ , so $maj ([2, 0, 3, 1]) = 1 + 3 = 4$ . The image $[2, 3, 0, 1]$ has inversions $(2, 0)$ , $(2, 1)$ , $(3, 0)$ , and $(3, 1)$ , so $inv ([2, 3, 0, 1]) = 4$ . This matches the theorem.

step 1.1givendischarge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.

Foata's transformation on the permutation 2 0 3 12031