TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Krawczyk inclusion gives solutions and contraction gives uniqueness

Statement

Let $n \geq 1$ , let $A \subseteq C^{n \times n}$ and $b \subseteq C^{n}$ be nonempty, let $X$ be a nonempty closed complex rectangle box in $C^{n} ≅ R^{2 n}$ , let $x_{0} \in X$ , and let the point matrix $R$ be nonsingular. For each fixed $A \in A, b \in b$ , define

Φ_{A, b} (x) = x - R (A x - b)

and

K^{*} (X) = {x_{0} - R (A x_{0} - b) + (I - R A) (x - x_{0}) : A \in A, b \in b, x \in X} .

If $K^{*} (X) \subseteq X$ , then every admitted pair $(A, b)$ has a solution in $X$ . If in addition a number $q < 1$ satisfies $∥ I - R A ∥_{\infty, C} \leq q$ for every $A \in A$ , that solution is unique in $X$ for each fixed pair.

Facts & Assumptions

Given: The families, box, center, and nonsingular $R$ in the statement.

[F1]

Every continuous self-map of a closed box has a fixed point (Brouwer fixed points exist on every nonempty closed box).

[F2]

The complex infinity matrix norm is compatible with multiplication by vectors (The complex infinity matrix norm is compatible and submultiplicative).

[F3]

The vector infinity norm is the maximum coordinate modulus (The complex induced infinity matrix norm).

Proof

1.1givenF1algebra

Fix any $A \in A, b \in b$ . Algebra gives $Φ_{A, b} (x) = x_{0} - R (A x_{0} - b) + (I - R A) (x - x_{0})$ , so $K^{*} (X) \subseteq X$ implies $Φ_{A, b} (X) \subseteq X$ . The affine map $Φ_{A, b}$ is continuous on the closed box, so [F1] gives a fixed point $x \in X$ . Since $R$ is nonsingular, $Φ_{A, b} (x) = x$ is equivalent to $A x = b$ .

2.1givenF2F3step 1.1∎

If $x, y \in X$ both solve the same system, then both are fixed points and $x - y = (I - R A) (x - y)$ . By [F2] and [F3], $∥ x - y ∥_{\infty} \leq q ∥ x - y ∥_{\infty}$ . Since the vector infinity norm [F3] vanishes only at the zero vector and $q < 1$ , $x = y$ . This proves uniqueness for that pair; the pair was arbitrary.

Depends on

Used by

Dependency tree · two levels

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Sources

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