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.

Every local minimum of a convex function on an open Euclidean convex set is global

Statement

Let $f : U \to R$ be convex on an open convex set. Every local minimizer of $f$ (Local and strict local extrema for scalar fields on Euclidean open sets) is a global minimizer.

Facts & Assumptions

Given: A local minimizer $a \in U$ of the function in the Statement.

[F1]

The function $f : U \to R$ is convex when $f ((1 - t) x + t y) \leq (1 - t) f (x) + t f (y)$ for all $x, y \in U$ and $t \in [0, 1]$ (Convex and strictly convex functions on Euclidean convex sets).

Proof

technique · contradiction

1.1

Suppose $a$ were not a global minimizer, and choose $y \in U$ with $f (y) < f (a)$ . For sufficiently small $0 < t < 1$ , the point $z = (1 - t) a + t y$ lies in the local-minimum neighbourhood of $a$ , while [F1] gives $f (z) \leq (1 - t) f (a) + t f (y) < f (a),$ a contradiction.

F1assume-contraalgebra

2.1

The assumption in step 1.1 is untenable, so no domain point has value below $f (a)$ and the local minimizer is global.

step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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 5 (standard reference, not scraped)
S. Boyd and L. Vandenberghe, Convex Optimization, §3.1 (standard reference, not scraped)