PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Length dominates endpoint distance

Statement

For every piecewise $C^{1}$ curve $γ : [a, b] \to M$ , $d_{g} (γ (a), γ (b)) \leq L_{g} (γ)$ .

Facts & Assumptions

Given: A connected Riemannian manifold and a piecewise $C^{1}$ curve.

[F1]

Riemannian distance on a connected manifold: On a connected Riemannian manifold define $d_{g} (p, q) = \inf {L_{g} (γ) : γ is piecewise C^{1} from p to q}$ . Lengths are those of def-riemannian-speed-and-length. For each pair $p, q$ , lem-any-two-points-in-a-connected-smooth-manifold-can-be-joined-by-a-piecewise-c-one-curve supplies a curve, so the set of lengths is nonempty, contains a finite real number and is bounded below by zero. Applying the least-upper-bound property cor-cauchy-reals-lub-complete to the negatives gives a finite nonnegative infimum. On the empty connected manifold this defines the empty distance function; there are no pairs to evaluate. No minimizing curve is part of this definition.

Proof

technique · direct

1.1

The curve $γ$ itself belongs to the family defining $d_{g} (γ (a), γ (b))$ , so its finite length is an element of that nonempty set.

F1given

2.1

An infimum is a lower bound for every element of its defining set. Applying this to $L_{g} (γ)$ proves the inequality, including a constant curve or singleton interval, when both sides are zero.

F1step 1.1∎

Source locator

Lee, Chapter 13, pp.337–340, Proposition 13.25, Lemma 13.28 and Theorem 13.29; finite piecewise $C^{1}$ refinements and pauses are treated explicitly here.

Depends on

Riemannian distance on a connected manifold

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)