ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-30

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.

Sparse Gaussian elimination can create fill-in in the factors

Example

The sparse matrix $A = (\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array})$ has a zero in position $(3, 2)$ , but after the first elimination step that entry becomes $- 1$ . Thus Gaussian elimination can create nonzeros that were absent in the original matrix.

Facts & Assumptions

Given: The displayed sparse matrix $A$ .

[L1]

Pivoting language records the elimination process entry by entry (Permutation matrices, partial pivoting, and the pivot-growth factor).

Verification

technique · direct

1.1

Use the first pivot $a_{11} = 1$ . Eliminating the entries below it subtracts row $1$ from rows $2$ and $3$ , giving $(\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 1 \end{array}) .$ The entry in position $(3, 2)$ was $0$ before elimination and is now $- 1$ .

givenL1algebra

2.1

The new nonzero in step 1.1 is fill-in: it appears after one elimination step even though the corresponding original entry was zero.

step 1.1

3.1

Steps 1.1-2.1 verify the example.

step 1.1step 2.1∎

Depends on

Permutation matrices, partial pivoting, and the pivot-growth factor

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

David Bindel, CS 4220: Numerical Analysis, Blocked LU and Cholesky (standard reference, not scraped)