Plain-English statement
A metric curve has finite instantaneous speed at almost every time.
Mathematical statement
for almost every t there is speed>=0 such that dist(mu_s,mu_t)/abs(s-t) tends to speed as s tends to t with s not equal to t.
Intuition
Distance quotients retain the magnitude of velocity even when the ambient space has no vector subtraction.
Conditions
- a pseudometric state space
- real time
- Lebesgue almost-everywhere existence of the limit
Why these conditions cannot be dropped
- puncturing removes division by zero
- the a.e. condition matches the source definition
Proof route
- define HasMetricDerivativeAt
- quantify its finite real witness almost everywhere
Lean interface notes
- uses nhdsWithin t {t} complement through nhdsNE
- this is the source's informal criterion, not the upper-gradient refinement
Read the mathematics first, then descend into Lean
You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.
Proof architecture
Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.
Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.
How to read the exact Lean declaration
Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.
- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.
Source voice and ASTIS voice stay separate
When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.
Lean statement
def IsAbsolutelyContinuousMetricCurve
{M : Type*} [PseudoMetricSpace M]
(curve : ℝ → M) : Prop :=
∀ᵐ t ∂(by volume_tac : Measure ℝ),
∃ speed : ℝ, HasMetricDerivativeAt curve speed t
end MetricCurve
end Geometry
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/Geometry/MetricCurve.lean:34published source at 7bcd37294df1