LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-13

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.

Refining a partition cannot decrease its inscribed polygonal length

Statement

Let $γ : [a, b] \to R^{n}$ be a path with $a < b$ and $n \geq 1$ . If a partition $Q$ refines a partition $P$ , then

$ℓ_{P} (γ) \leq ℓ_{Q} (γ) .$

Facts & Assumptions

Given: A path and partitions $P \subseteq Q$ in the refinement sense.

[L1]

A refinement contains every point of the original partition, possibly with additional points between consecutive old points (Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions).

[L2]

The Euclidean norm satisfies the triangle inequality (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

Proof

technique · direct

1.1L2

If one point $s$ is inserted between consecutive points $u < v$ of $P$ , [L2] gives $∥ γ (v) - γ (u) ∥_{2} \leq ∥ γ (s) - γ (u) ∥_{2} + ∥ γ (v) - γ (s) ∥_{2}$ .

2.1step 1.1

Every other summand is unchanged, so insertion of one point cannot decrease polygonal length.

3.1givenL1step 2.1∎

Because $Q$ is finite and contains $P$ , it is obtained by finitely many one-point insertions. Repeated use of step 2.1 gives $ℓ_{P} (γ) \leq ℓ_{Q} (γ)$ .

Depends on

Dependency tree · 2 levels

Sources

A. R. Shastri, Metric Spaces, Section 5