ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Two explicit $2$ by $2$ matrices do not commute

Example

In $M_2(F)$ , let

$A=E_{01}=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad B=E_{10}=\begin{pmatrix}0&0\\1&0\end{pmatrix}.$

Then $AB\ne BA$ .

Facts & Assumptions

Given: A field $F$ and the displayed matrix units in $M_2(F)$ .

[L1]

Matrix units satisfy $E_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}$ ( $E_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}$ ).

Verification

technique · direct

1.1

By [L1], $AB=E_{00}=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ while $BA=E_{11}=\begin{pmatrix}0&0\\0&1\end{pmatrix}$ .

givenL1

2.1

The products differ at entry $(0,0)$ , since $1\ne0$ in a field, so $A$ and $B$ do not commute.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Two explicit 222 by 222 matrices do not commute