TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ ,

$tr (A B) = tr (B A) .$

The two products may have different sizes, and the equality includes $m = 0$ or $n = 0$ .

Given: A field $F$ and the rectangular matrices in the Statement.

[L1]

[L2]

technique · direct

1.1givenL1

Expanding by [L1] gives $tr (A B) = \sum_{i < m} \sum_{j < n} a_{i j} b_{j i}$ .

2.1step 1.1L1L2

By [L2] and commutativity of multiplication in $F$ , this equals $\sum_{j < n} \sum_{i < m} b_{j i} a_{i j}$ .

3.1step 2.1L1L2∎

The final double sum is $tr (B A)$ by [L1]. If $m = 0$ or $n = 0$ , both sides are empty double sums and equal $0$ .

Depends on