Plain-English statement
- Named coordinate-divergence version of the finite Euclidean Gibbs-weighted first-derivative display. The vector field is the coordinate representative `y ↦ exp (-V y) * fderiv ℝ f y eᵢ`. The theorem only rewrites the compiled coordinate-sum display through the local pointwise `coordinateDivergence` definition. It does not identify this with any integration theorem, prove weighted integration by parts, establish generator domains, or prove invariant Gibbs law, reversibility, or KL/FI dissipation.
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
theorem coordinateDivergence_expNeg_fderivCoordinateField_langevinGenerator_display_of_differentiableAt
{ι : Type*} [Fintype ι] [DecidableEq ι]
{V f : EuclideanSpace ℝ ι → ℝ} {x : EuclideanSpace ℝ ι}
(hV : DifferentiableAt ℝ V x)
(hfderiv : DifferentiableAt ℝ
(fun y : EuclideanSpace ℝ ι => fderiv ℝ f y) x)
(hf : DifferentiableAt ℝ f x) :
_root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence
(fun y : EuclideanSpace ℝ ι =>
(WithLp.toLp 2 (fun i =>
Real.exp (-V y) * fderiv ℝ f y (EuclideanSpace.single i (1 : ℝ))) :
EuclideanSpace ℝ ι)) x =
Real.exp (-V x) * (Laplacian.laplacian f x -
inner ℝ (gradient V x) (gradient f x)) := by
dsimp [
_root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence]
simpa using
finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display_of_differentiableAt
(V := V) (f := f) (x := x) hV hfderiv hf
/-- Trace-summand display for the explicit Pi-space vector field
`x ↦ exp (-V x) * fderiv f x eᵢ`.
This is the pointwise bridge from Mathlib's finite-box divergence-theorem trace
integrand to the Langevin display
`exp (-V) * (Δ f - <∇V, ∇f>)`. It uses the supplied Frechet derivative of the
explicit Pi-space vector field, plus pointwise differentiability assumptions
needed by the compiled coordinate-divergence display.
It does not prove that the supplied derivative exists on a box, prove
continuity or integrability, invoke a divergence theorem, cancel boundary
terms, establish weighted integration by parts, define a generator domain, or
prove invariant/reversible Gibbs laws. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:549published source at 7bcd37294df1
Proof architecture
Chewi Ch.1 Langevin root: rewrite the explicit Gibbs-weighted first-derivative coordinate field through ASTIS `coordinateDivergence` and recover `exp(-V x) * (Delta f - <grad V, grad f>)` pointwise
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `simpa` closes the goal after a controlled simplification of a typed result.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Totalized `fderiv` values must not be read as a differentiability theorem.
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.
- A Gibbs expression is not automatically a probability law or an invariant law.
- A formal generator display does not establish a closed operator domain.