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TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass
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The interval determinant sign certifier is sound and failure means unknown

Statement

If Outward Leibniz determinant sign and nonsingularity certifier returns nonsingular or all-singular for a real interval matrix family, the corresponding nonsingularity/sign or all-singular claim holds for every family member. An inconclusive result makes no claim about singularity.

Facts & Assumptions

Given: A nonempty real square matrix family represented by finite interval boxes and a completed algorithm return.

[F1]

When every operation returns a finite enclosure, outward interval DAG evaluation encloses the exact determinant range; a non-finite status asserts no finite DAG enclosure (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F2]

An interval containing all determinants certifies nonsingularity when it excludes zero; the exact zero interval certifies singularity (An interval determinant excluding zero certifies nonsingularity and sign).

[F3]

The determinant algorithm evaluates the finite Leibniz formula and returns only the stated three statuses (Outward Leibniz determinant sign and nonsingularity certifier).

Proof

1.1givenF1F3algebra

Suppose the return is nonsingular or all-singular. By [F3], every operation in the finite Leibniz evaluation returned a finite enclosure, so [F1] gives det⁡A∈D for every represented family member. If the return is nonsingular, then 0∉D; if it is all-singular, then D={0}.

2.1F1F2F3step 1.1∎

Apply [F2] to D in each of those two conclusive cases to obtain the claimed singularity or determinant-sign guarantee. By [F3], an inconclusive return occurs either after a non-finite operation, when [F1] asserts no finite determinant enclosure, or after a finite enclosure with 0∈D and D≠{0}, when neither test in [F2] applies. In either branch the algorithm makes no singularity or sign claim.

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Outward Leibniz determinant sign and nonsingularity certifier.

Dependency tree · two levels

12 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