ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

Natural interval subtraction loses repeated-input correlation

Example

Let the single real input $x$ range over $X = [1, 2]$ , and evaluate the expression DAG $f (x) = x - x$ by its natural interval extension. The exact range is $f (X) = {0}$ , while the interval computation returns $X - X = [- 1, 1]$ . Both endpoints are sound, but the output is wider because the two occurrences of $x$ are evaluated as independent members of $X$ .

Facts & Assumptions

Given: The real interval $X = [1, 2]$ , the expression $f (x) = x - x$ , and the outward subtraction rule.

[L1]

Outward-rounded interval, rectangle and ball operations: for $[a, b] - [c, d]$ , the interval result before final endpoint rounding is $[a - d, b - c]$ .

[L2]

Outward-rounded operations are inclusion-isotone and enclose DAG evaluations: natural interval evaluation encloses every exact DAG value.

Verification

1.1givenL1algebra

Substituting $a = c = 1$ and $b = d = 2$ into [L1] gives $[1, 2] - [1, 2] = [1 - 2, 2 - 1] = [- 1, 1]$ .

2.1givenL2algebra∎

For every real $x \in [1, 2]$ , the two DAG nodes have the same input value, so $f (x) = x - x = 0$ , and at $x = 1$ this value is attained. Thus the exact range is ${0}$ , which is contained in the computed interval by [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Kearfott, Interval Computations: Introduction, Uses, and Resources, §1 (standard reference, not scraped)