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.

Axis-parallel rectangles in $R^{m}$ and their volume

Definition

Fix a natural number $m \geq 1$ . For $a, b \in R^{m}$ with $a_{j} \leq b_{j}$ for $j < m$ , define $[a, b] : = {x \in R^{m} : a_{j} \leq x_{j} \leq b_{j} (j < m)}, vol [a, b] : = \prod_{j < m} (b_{j} - a_{j}) .$ The product is the recursively defined finite product of Finite sums and finite products, by recursion. The rectangle is nondegenerate when every $a_{j} < b_{j}$ , and it is a cube when all side lengths are equal.

Every factor is nonnegative, so volume is nonnegative. For a coordinate index $r < m$ , cutting at $c \in [a_{r}, b_{r}]$ gives two rectangles whose volumes add to the original, by distributivity in that factor and Laws of finite sums and finite products. Under the standard identification $R^{1} ≅ R$ ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it, The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ ), this is the interval $[a_{0}, b_{0}]$ and its length.

Axis-parallel rectangles in $R^{m}$ and their volume

Definition

Depends on

Dependency tree · 2 levels

Sources

Axis-parallel rectangles in RmRm and their volume

Definition

Depends on

Dependency tree · 2 levels

Sources

Axis-parallel rectangles in $R^{m}$ and their volume