ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)audited 2026-08-02

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 absolute-value function is convex

Example

The function $f(x)=|x|$ is convex on $\mathbb R$ . Indeed, for $0\le\lambda\le1$ ,

$|\lambda x+(1-\lambda)y|\le|\lambda x|+|(1-\lambda)y|=\lambda|x|+(1-\lambda)|y|,$

by the triangle inequality (The triangle inequality) and absolute homogeneity (Basic properties of the absolute value), which is precisely the convexity inequality (Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval).

Remarks

Convexity does not entail differentiability at every point; the nondifferentiability of absolute value at zero is recorded in $x \mapsto |x|$ is continuous everywhere and not differentiable at $0$ : the difference quotient equals $1$ on the right and $-1$ on the left, so the two one-sided limits differ but is not a dependency of this example.

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: 14 results over 6 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

S. Boyd and L. Vandenberghe, Convex Optimization, §3.1 (standard reference, not scraped)