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.

Resolvent of a monotone relation

Definition

For a relation $A \subseteq E \times E$ and $ρ > 0$ , define its resolvent as the inverse relation $J_{ρ A} : = (I + ρ A)^{- 1}, J_{ρ A} (y) = {x \in E : y - x \in ρ A (x)},$ where $ρ A (x) = {ρ a : a \in A (x)}$ . Its domain is the set of $y$ for which this value set is nonempty. At this point it may be partial and set-valued; full domain and single-valuedness are separate conclusions of A maximal-monotone resolvent is everywhere-defined and firmly nonexpansive.

Facts & Assumptions

Given: A relation $A$ on $E$ and a scalar $ρ > 0$ .

[A1]

The relation is on $E$ and $ρ > 0$ .

[F1]

Monotone and maximal-monotone relations on Euclidean space: $A (x)$ denotes the set of $a$ with $(x, a) \in A$ .

Proof

1.1A1givenF1given

The equation $y - x \in ρ A (x)$ means that some $a \in A (x)$ satisfies $y = x + ρ a$ ; this is precisely the inverse graph of $I + ρ A$ .

2.1A1givenF1given∎

The value $J_{ρ A} (y)$ is a set and may be empty until a theorem proves existence; the definition itself asserts no surjectivity or uniqueness.

Depends on

Monotone and maximal-monotone relations on Euclidean space

Used by

Dependency tree · two levels

2 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)