ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Gauges of norm balls and finite-dimensional ellipsoids

Example

For $B (0, r) = {x : ∥ x ∥ < r}$ , $p_{B} (x) = ∥ x ∥ / r$ . For a positive-definite real quadratic form $q$ on $R^{n}$ and $E = {x : q (x) < 1}$ , $p_{E} (x) = \sqrt{q (x)}$ . Both are norms: the first is a positive scalar multiple of the given norm, and the second is the norm induced by the inner product associated with the positive-definite quadratic form $q$ .

Facts & Assumptions

Given: A radius $r > 0$ and a positive-definite quadratic form $q$ .

[F1]

The gauge of an absorbing set is the infimum of its admissible positive dilates (Minkowski functional of an absorbing set).

Verification

technique · direct

1.1

$x \in t B (0, r)$ exactly when $∥ x ∥ < t r$ , whose infimum over $t > 0$ is $∥ x ∥ / r$ .

F1givenalgebra

2.1

Likewise $x \in t E$ exactly when $q (x) < t^{2}$ , so the infimum is $\sqrt{q (x)}$ . Positive definiteness gives zero only at $x = 0$ , and the displayed formulas give norm homogeneity and triangle inequality.

F1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Gerald Teschl, Topics in Real and Functional Analysis, §5.1 examples (standard reference, not scraped)