Alphabeta MathHow 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 is convex on . Indeed, for ,
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 is continuous everywhere and not differentiable at : the difference quotient equals on the right and 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 · two levels
9 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
- S. Boyd and L. Vandenberghe, Convex Optimization, §3.1 (standard reference, not scraped)