TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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 interval Krawczyk certifier has the stated per-system guarantee

Statement

If Interval Krawczyk linear system certifier with explicit statuses returns existence-certified, every admitted system $A x = b$ has at least one solution in $X$ . If it also returns unique-certified, each fixed admitted pair has exactly one solution in $X$ . Inconclusive asserts neither existence nor singularity.

Facts & Assumptions

Given: The admitted matrix and right-hand-side families, box $X$ , $x_{0} \in X$ , a supplied inverse witness $S$ for $R$ , and an existence-certified run.

[F1]

Outward interval operations enclose their exact expression ranges (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F2]

Krawczyk set inclusion gives existence, and a defect norm below one gives uniqueness (Krawczyk inclusion gives solutions and contraction gives uniqueness).

[F3]

On an existence-certified return the admitted inputs satisfy $R S = S R = I$ , and the algorithm checks that its outward Krawczyk enclosure lies in $X$ ; on a unique-certified return it also supplies an outward defect bound $q < 1$ (Interval Krawczyk linear system certifier with explicit statuses).

Proof

1.1givenF1F2F3algebra

Since $R S = S R = I$ , the supplied $R$ is invertible. By [F1], the computed enclosing box contains every exact value of $x_{0} - R (A x_{0} - b) + (I - R A) (x - x_{0})$ for each admitted pair and each $x \in X$ . Its successful inclusion in $X$ therefore implies the set $K^{*} (X)$ from [F2] is contained in $X$ .

2.1givenF2F3step 1.1∎

Apply [F2] to each fixed pair to obtain a solution in $X$ . If the run also reports unique-certified, [F3] supplies an outward row-sum bound with $∥ I - R A ∥_{\infty, C} \leq q < 1$ for every admitted $A$ , so the uniqueness clause of [F2] applies separately to every pair. If that strict bound is not returned, no uniqueness claim follows.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Siegfried M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010), §§10.5–10.6 (standard reference, not scraped)