LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate

Statement

If $P^{'}$ refines a grid $P$ , then $L (f, P) \leq L (f, P^{'}) \leq U (f, P^{'}) \leq U (f, P)$ . Moreover, for a fixed grid $P$ , there is a constant $C_{P}$ such that refining any grid of mesh $δ$ by $P$ changes either Darboux sum by at most $2 B C_{P} δ$ , where $∣ f ∣ \leq B$ .

Facts & Assumptions

Given: The grids, bounded $f$ , and bound $B$ .

[L1]

Darboux sums and iterated cell sums are Lower and upper Darboux sums over a grid partition in $R^{m}$ and Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh.

[L2]

Finite sums split and multiplication distributes (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

Proof

technique · induction

1.1baseL1L2

Insert one coordinate hyperplane. Every new cell lies in one old cell, so its infimum is no smaller and its supremum no larger. Splitting the affected coordinate sum proves the four inequalities.

1.2L1L2given

Only fine cells meeting an interior hyperplane of $P$ can cross a coarse-cell boundary. For a hyperplane perpendicular to coordinate $j$ , those cells lie in a slab of thickness at most $2 δ$ ; repeated product distributivity bounds their total volume by $2 δ \prod_{r \neq j} (b_{r} - a_{r})$ .

2.1ihstep 1.1given

Iterating over the finitely many inserted hyperplanes and coordinates proves refinement monotonicity.

2.2step 1.2L2given

Sum this bound over the finitely many fixed interior hyperplanes to define $C_{P}$ . On all other cells refinement changes no coarse bound, while on boundary cells each contribution changes by at most $2 B$ times its volume.

Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources