DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Monotone and maximal-monotone relations on Euclidean space

Definition

Fix $n \in N$ and the Euclidean space $E = R^{n}$ with inner product and norm from The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . A relation $A$ on $E$ is a subset of $E \times E$ ; write $a \in A (x)$ exactly when $(x, a) \in A$ . Its graph is this subset. The relation is monotone if $⟨ x - x^{'}, a - a^{'} ⟩ \geq 0 for all (x, a), (x^{'}, a^{'}) \in A .$ It is maximal monotone if it is monotone and no strictly larger subset of $E \times E$ is monotone. This definition is finite-dimensional and makes no claim about infinite-dimensional spaces.

Facts & Assumptions

Given: $n \in N$ and $E = R^{n}$ .

[A1]

The ambient space is $E = R^{n}$ and $A \subseteq E \times E$ .

[F1]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $⟨ x, y ⟩ = \sum_{i < n} x_{i} y_{i}$ is a real bilinear inner product on $E$ .

Proof

1.1A1givenF1given

A relation is a graph of ordered pairs $(x, a)$ with both coordinates in $E$ ; the notation $a \in A (x)$ is the same membership statement. The displayed inequality therefore quantifies over two graph points and is well-defined by [F1].

2.1A1givenF1given∎

Maximality is inclusion-maximality among monotone graphs: it rules out a monotone proper extension and does not mean that the graph is a largest monotone subset. These are the graph-level conventions used below.

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

Resolvent of a monotone relation Definition
A maximal-monotone resolvent is everywhere-defined and firmly nonexpansive Theorem

Dependency tree · two levels

14 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

Jonathan Eckstein and Dimitri P. Bertsekas, On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, §§2–4 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, §4.4 (standard reference, not scraped)