DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-01

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.

Lower and upper Darboux sums over a grid partition in $R^{m}$

Definition

Let $f : Q \to R$ be bounded on a nondegenerate rectangle and let $P$ be a grid. For each cell $Q_{i}$ , put $m_{i} : = \inf f [Q_{i}], M_{i} : = \sup f [Q_{i}], L (f, P) : = \sum_{i} m_{i} vol (Q_{i}), U (f, P) : = \sum_{i} M_{i} vol (Q_{i}) .$ The extrema exist as finite reals because each nonempty image is bounded (Lower bound, bounded below, bounded set, Complete ordered field (least-upper-bound property), Every nonempty set bounded below has an infimum, Greatest lower bound (infimum), Suprema and infima are unique), and the sums use the iterated convention of Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh.

Since $m_{i} \leq M_{i}$ and cell volumes are nonnegative, $L (f, P) \leq U (f, P)$ . Moreover $U (f, P) - L (f, P) = \sum_{i} (M_{i} - m_{i}) vol (Q_{i}),$ the sum of cell oscillations weighted by volume (Laws of finite sums and finite products).

Lower and upper Darboux sums over a grid partition in $R^{m}$

Definition

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Dependency tree · 2 levels

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Lower and upper Darboux sums over a grid partition in RmRm

Definition

Depends on

Dependency tree · 2 levels

Sources

Lower and upper Darboux sums over a grid partition in $R^{m}$