LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Exterior Multiplication Koszul Sign Rule

Statement

If $i$ is distinct from $i_{1} < \dots < i_{p}$ , then $e_{i} \land e_{I} = (- 1)^{# {i_{j} < i}} e_{sort ({i} \cup I)}$ , while $e_{i} \land e_{I} = 0$ when $i \in I$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Exterior Algebra Of A Finite Free Module, Exterior Algebra Basis Monomials.

Proof

technique · direct

1.1givenalgebra

Move $e_{i}$ across precisely the factors indexed below $i$ ; each adjacent swap contributes $- 1$ .

2.1step 1.1algebra∎

If $i$ already occurs, the wedge contains a square and is zero; this remains true in characteristic $2$ .

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Koszul complexes and regular sequences