Plain-English statement
- Coordinate-unit version of `finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_basisHandoff`. This removes only the Gibbs-weight chain-rule hypothesis from the explicit coordinate-unit display. Divergence, coordinate product-rule, IBP, generator domains, stationarity, reversibility, invariant Gibbs law, and KL/FI dissipation remain separate red branches.
Read the mathematics first, then descend into Lean
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Proof architecture
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How to read the exact Lean declaration
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- 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
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Source voice and ASTIS voice stay separate
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Lean statement
theorem finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_coordinateHandoff
{ι : Type*} [Fintype ι] [DecidableEq ι]
{divWeighted : ℝ}
{V f : EuclideanSpace ℝ ι → ℝ} {x : EuclideanSpace ℝ ι}
{divCoord : ι → ℝ}
(hV : DifferentiableAt ℝ V x)
(hdivWeighted : divWeighted = ∑ i, divCoord i)
(hdiv : ∀ i,
divCoord i =
Real.exp (-V x) * iteratedFDeriv ℝ 2 f x
![(EuclideanSpace.single i (1 : ℝ)), (EuclideanSpace.single i (1 : ℝ))] +
(gradient (fun y : EuclideanSpace ℝ ι => Real.exp (-V y)) x) i *
(gradient f x) i) :
divWeighted =
Real.exp (-V x) * (Laplacian.laplacian f x -
inner ℝ (gradient V x) (gradient f x)) := by
exact finiteEuclidean_weightedDivergence_langevinGenerator_coordinateHandoff
(rho := Real.exp (-V x))
(divWeighted := divWeighted)
(V := V) (f := f) (x := x)
(divCoord := divCoord)
(gradRho := fun i =>
(gradient (fun y : EuclideanSpace ℝ ι => Real.exp (-V y)) x) i)
hdivWeighted hdiv
(fun i =>
_root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_coordinate_eq_of_differentiableAt
(V := V) (x := x) hV i)
/-- Coordinate-line-derivative sum display for the explicit Gibbs-weighted
first-derivative field.
This theorem aggregates the compiled pointwise leaf
`lineDeriv_expNegPotential_mul_fderiv_coordinate_eq` across all finite
coordinates and then calls the existing Euclidean Langevin display handoff.
It removes the supplied coordinate product-rule/Hessian-diagonal hypothesis
from `finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_coordinateHandoff`
for the specific field
`x ↦ exp (-V x) * fderiv ℝ f x eᵢ`.
The hypothesis `hgradF` is intentionally still supplied: it is the pointwise
identification of the `fderiv` coordinate slice with Mathlib's `gradient`
coordinate. This theorem does not define a divergence operator, assert that
the displayed sum is the divergence of a vector field, prove integration by
parts, establish a semigroup-generator/domain theorem, or prove stationarity,
reversibility, invariant Gibbs law, or KL/FI dissipation. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:409published source at 7bcd37294df1
Proof architecture
Chewi SDE/DENS root: coordinate-unit handoff from supplied divergence sum and coordinate product-rule facts to `exp(-V x) * (Δ f - <∇V, ∇f>)`, with the Gibbs-weight chain-rule coordinate equality discharged from `DifferentiableAt ℝ V x`
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `exact` closes the current goal with an already typed term.
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
- 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.