ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)judge 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 $x,y\in B_2(c,r)\subseteq\mathbb R^n$ , with $0\le t\le1$ . The triangle inequality and absolute homogeneity give

$\lVert(1-t)x+ty-c\rVert_2\le(1-t)\lVert x-c\rVert_2+t\lVert y-c\rVert_2<r.$

So the segment from $x$ to $y$ stays in the ball. By A finite concatenation of straight segments in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ . 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 79 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Convex set (standard reference, not scraped)