ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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.

The biproduct of two abelian groups computed as a matrix calculus

Example

In $A b$ , the endomorphism

$f : Z \oplus Z \to Z \oplus Z, f (x, y) = (2 x - y, 3 x),$

has matrix

$(\begin{array}{cc} 2 & - 1 \\ 3 & 0 \end{array}),$

and if

$g (x, y) = (x + 4 y, - y),$

then the matrix of $g \circ f$ is the product

$(\begin{array}{cc} 1 & 4 \\ 0 & - 1 \end{array}) (\begin{array}{cc} 2 & - 1 \\ 3 & 0 \end{array}) = (\begin{array}{cc} 14 & - 1 \\ - 3 & 0 \end{array}) .$

Facts & Assumptions

Given: The biproduct $Z \oplus Z$ in $A b$ and the two displayed homomorphisms $f$ and $g$ .

[L1]

Morphisms between finite biproducts correspond to matrices of their component maps (Morphisms between finite biproducts correspond to matrices).

[L2]

Composition of such morphisms is matrix multiplication (Composition of morphisms between finite biproducts is matrix multiplication).

Verification

technique · direct

1.1

The four component maps of $f$ are $x \mapsto 2 x$ , $y \mapsto - y$ , $x \mapsto 3 x$ , and $y \mapsto 0$ , so [L1] gives the displayed matrix for $f$ . Likewise $g$ has matrix $(\begin{array}{cc} 1 & 4 \\ 0 & - 1 \end{array})$ .

2.1

Applying [L2] yields the matrix product $(\begin{array}{cc} 14 & - 1 \\ - 3 & 0 \end{array})$ for $g \circ f$ . Evaluating the composite directly gives $(g \circ f) (x, y) = (14 x - y, - 3 x)$ , which matches the same matrix.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.2 (standard reference, not scraped)