LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

An odd permutation reverses the sign of an oriented simplex

Statement

If $π \in S_{n + 1}$ is odd, then $[v_{π (0)}, \dots, v_{π (n)}] = - [v_{0}, \dots, v_{n}]$ as oriented simplices.

Proof

Given: An odd permutation $π$ of the vertices of an $n$ -simplex.

1.1

Every odd permutation is a product of an odd number of transpositions, and each transposition switches the two orientation classes by definition.

given

2.1

After composing an odd number of such sign reversals, the final ordering represents the opposite orientation class, so its oriented simplex is the negative of the original one.

step 1.1∎

Depends on

An orientation of a simplex

Used by

The simplicial boundary is independent of the chosen oriented representative Lemma

Dependency tree · two levels

2 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.

Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)