CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Zero is a subgradient exactly at a global minimum

Statement

Let $f : C \to R$ be convex and let $a \in C$ . Then $0 \in \partial f (a)$ if and only if $f (a) \leq f (y)$ for every $y \in C$ .

Facts & Assumptions

Given: The function and point in the Statement.

[F1]

A vector $v$ is a subgradient of $f$ at $a$ when $f (y) \geq f (a) + ⟨ v, y - a ⟩$ for every $y$ in the domain (Subgradients and the subdifferential of a convex function).

Proof

technique · direct

1.1

For the forward implication, put $v = 0$ in [F1]. The result is $f (y) \geq f (a)$ for every $y \in C$ , exactly the global-minimum condition.

F1algebra

2.1

For the reverse implication, if $a$ is a global minimizer then $f (y) \geq f (a) = f (a) + ⟨ 0, y - a ⟩$ for every $y$ . This is [F1] with $v = 0$ , so $0 \in \partial f (a)$ .

F1assume-hyp∎

Depends on

Subgradients and the subdifferential of a convex function

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lecture 12 (standard reference, not scraped)
D. Drusvyatskiy, Convex Analysis and Nonsmooth Optimization, Corollary 3.35 (standard reference, not scraped)