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

Soundness of finite erfcx enclosures

Statement

If Finite certified evaluation of the scaled complementary error function returns success for x>0, the returned interval contains erfcx⁡(x) and passes the requested absolute-radius test. A relative certificate is valid only if zero is outside the interval and the required relative-radius test passes. Every execution with finite caps terminates. The theorem makes no success claim for every x or finite cap.

Facts & Assumptions

Given: A successful return of Finite certified evaluation of the scaled complementary error function using a sound outward interval backend.

[A1]

Certified range evaluation for elementary real and complex functions supplies the finite square-root range call for π; soundness of that real output is given by Certified real elementary-function evaluations enclose their ranges.

[A2]

Outward-rounded interval, rectangle and ball operations and Outward-rounded operations are inclusion-isotone and enclose DAG evaluations: every finite arithmetic interval operation encloses its exact result.

[F1]

Finite continued-fraction enclosure for erfcx: each finite-depth continued- fraction interval contains the exact value.

[F2]

Signed finite remainder for the erfcx asymptotic expansion: the signed remainder interval formed from SN and BN contains the exact value for every N≥1.

[F3]

Certified special-function enclosure result: success requires a containing interval with the stated radius, and relative certification requires zero exclusion.

[F4]

Finite certified evaluation of the scaled complementary error function: the procedure returns only a continued-fraction or signed-asymptotic candidate, or a non-success status.

Proof

1.1A1A2F1F2F4

A continued-fraction candidate contains erfcx⁡(x) by [F1]. A signed-asymptotic candidate contains it by [F2]. The interval evaluation of each candidate is sound by [A1] and [A2]. If the procedure intersects both candidates, their common exact value lies in the intersection; otherwise the selected candidate remains an enclosure.

2.1A2F4step 1.1

Thus every interval eligible for a success return contains the exact value. The finite arithmetic and intersection preserve inclusion by [A2] and the algorithm definition.

3.1F3F4step 2.1

The procedure reports success only after the absolute-radius test in [F3] and sets a relative flag only when the interval excludes zero and the relative-radius test passes. No width decrease or universal success is asserted.

4.1givenA1step 2.1step 3.1algebra∎

The stage loop is finite; every continued-fraction recurrence, asymptotic sum, and elementary-range call has finite depth, terms, format, and truncation index. Every branch has an explicit return or proceeds to the next bounded stage. Therefore every capped execution terminates.

Depends on

Used by

Cited to discharge well-definedness by Finite certified evaluation of the scaled complementary error function.

Dependency tree · two levels

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Sources