CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

For a C1 path the arc-length accumulation function has derivative equal to speed

Let $a < b$ , let $γ : [a, b] \to R^{n}$ be $C^{1}$ , and let

$s_{γ} (t) : = L_{[a, t]} (γ ∣_{[a, t]}) .$

Then $s_{γ}$ is differentiable on $[a, b]$ in the relative sense and

$s_{γ}^{'} (t) = ∥ γ^{'} (t) ∥_{2} .$

At $a$ and $b$ these are the relative one-sided derivatives.

Given: The $C^{1}$ path in the Statement.

[L1]

The arc-length function is $s_{γ} (t) = L_{[a, t]} (γ ∣_{[a, t]})$ , with $s_{γ} (a) = 0$ (The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path).

[L2]

[L3]

The integral function of an integrable function is differentiable at every point where the integrand is continuous, with derivative equal to the integrand; at endpoints this means the relative one-sided derivative (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

technique · direct

1.1givenalgebra

The speed $v (t) : = ∥ γ^{'} (t) ∥_{2}$ is continuous because $γ^{'}$ and the Euclidean norm are continuous.

2.1L1L2step 1.1

By [L1] and [L2], for every $t \in [a, b]$ , $s_{γ} (t) = \int_{a}^{t} v (u) d u,$ including $t = a$ .

3.1step 1.1step 2.1L3∎

Apply [L3] to step 2.1. It gives $s_{γ}^{'} (t) = v (t) = ∥ γ^{'} (t) ∥_{2}$ throughout $[a, b]$ , with the asserted endpoint interpretation.

Depends on