ExampleConstruction: Literature-sourcedVerification: 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.

A full $3 \times 3$ Leibniz expansion lists all six permutations and their signs

Example

For $A = (a_{i j}) \in M_{3} (R)$ , $\det A = a_{00} a_{11} a_{22} + a_{10} a_{21} a_{02} + a_{20} a_{01} a_{12} - a_{10} a_{01} a_{22} - a_{20} a_{11} a_{02} - a_{00} a_{21} a_{12} .$

Facts & Assumptions

Given: A $3 \times 3$ matrix over a commutative ring.

[L1]

The determinant is the signed Leibniz sum (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L2]

A permutation's sign is determined by the parity of its inversions (Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations).

[L3]

A set of size $3$ has $3! = 6$ bijections to itself (A finite set $A$ with $∣ A ∣ = n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality).

Verification

technique · direct

1.1

The even permutations in one-line notation are $[0, 1, 2]$ , $[1, 2, 0]$ and $[2, 0, 1]$ ; the odd ones are $[1, 0, 2]$ , $[2, 1, 0]$ and $[0, 2, 1]$ . These are all six permutations.

L1L2L3

2.1

Substitution yields the displayed six terms. For $A = (\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{array})$ they give $0 + 0 + 40 - 0 - 15 - 24 = 1$ , a concrete check of the signs.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Example 4.27 (standard reference, not scraped)
P. Massot, Structures algébriques fondamentales, §6.4 (standard reference, not scraped)

A full 3×33×3 Leibniz expansion lists all six permutations and their signs