ExampleConstruction: Literature-sourcedVerification: AI-generatedjudge pass (z-ai/glm-5.2)audited 2026-07-31

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 straight segment between two points of an open Euclidean ball stays in the ball

Example

Let $x, y \in B_{2} (c, r) \subseteq R^{n}$ , with $0 \leq t \leq 1$ . The triangle inequality and absolute homogeneity give

$∥ (1 - t) x + t y - c ∥_{2} \leq (1 - t) ∥ x - c ∥_{2} + t ∥ y - c ∥_{2} < r .$

So the segment from $x$ to $y$ stays in the ball. By A finite concatenation of straight segments in $R^{n}$ is a continuous path it is a polygonal path, and $B_{2} (c, r)$ is polygonally connected in the sense of Polygonal paths and polygonally connected subsets of $R^{n}$ . The norm and ball conventions are those of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and Open ball, closed ball and sphere in a metric space.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

Convex set (standard reference, not scraped)