Plain-English statement
- Supplied-hypothesis finite-coordinate handoff from weighted-divergence algebra to the Mathlib pointwise expression `Δ f - <∇V, ∇f>`. The hypotheses still provide the coordinate product-rule output, the Gibbs-weight chain-rule output, and the coordinate divergence sum. This theorem only replaces the coordinate second-derivative and gradient-product sums by Mathlib's `Laplacian.laplacian` and `gradient` display.
Read the mathematics first, then descend into Lean
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Proof architecture
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- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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- Proof actionsAfter
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Syntax used on this page
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Lean statement
theorem finiteEuclidean_weightedDivergence_langevinGenerator_basisHandoff
{ι : Type*} [Fintype ι]
{rho divWeighted : ℝ}
{V f : EuclideanSpace ℝ ι → ℝ} {x : EuclideanSpace ℝ ι}
{divCoord gradRho : ι → ℝ}
(hdivWeighted : divWeighted = ∑ i, divCoord i)
(hdiv : ∀ i,
divCoord i =
rho * iteratedFDeriv ℝ 2 f x
![(EuclideanSpace.basisFun ι ℝ) i, (EuclideanSpace.basisFun ι ℝ) i] +
gradRho i * (gradient f x) i)
(hgrad : ∀ i, gradRho i = -rho * (gradient V x) i) :
divWeighted =
rho * (Laplacian.laplacian f x -
inner ℝ (gradient V x) (gradient f x)) := by
have hcoord :
(∑ i, divCoord i) =
rho *
((∑ i, iteratedFDeriv ℝ 2 f x
![(EuclideanSpace.basisFun ι ℝ) i, (EuclideanSpace.basisFun ι ℝ) i]) -
∑ i, (gradient V x) i * (gradient f x) i) :=
finiteCoord_weightedDivergence_langevinGenerator_algebra
(rho := rho)
(divCoord := divCoord)
(hessDiag := fun i =>
iteratedFDeriv ℝ 2 f x
![(EuclideanSpace.basisFun ι ℝ) i, (EuclideanSpace.basisFun ι ℝ) i])
(gradRho := gradRho)
(gradV := fun i => (gradient V x) i)
(gradF := fun i => (gradient f x) i)
hdiv hgrad
have hdisplay := finiteEuclidean_langevinGenerator_basisDisplay V f x
calc
divWeighted = ∑ i, divCoord i := hdivWeighted
_ = rho *
((∑ i, iteratedFDeriv ℝ 2 f x
![(EuclideanSpace.basisFun ι ℝ) i, (EuclideanSpace.basisFun ι ℝ) i]) -
∑ i, (gradient V x) i * (gradient f x) i) := hcoord
_ = rho * (Laplacian.laplacian f x -
inner ℝ (gradient V x) (gradient f x)) := by rw [← hdisplay]
/-- Explicit coordinate-unit version of
`finiteEuclidean_weightedDivergence_langevinGenerator_basisHandoff`.
This is still a supplied-hypothesis algebra/display handoff. The theorem does
not prove the coordinate product rule, divergence theorem, integration by
parts, stationarity, reversibility, or any semigroup-generator statement. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:274published source at 7bcd37294df1
Proof architecture
Chewi SDE/DENS root: after coordinate product-rule and Gibbs-weight chain-rule facts are supplied, rewrite the finite weighted-divergence sum as `rho * (Laplacian.laplacian f x - inner ℝ (gradient V x) (gradient f x))` using the basis display
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
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.
- A Gibbs expression is not automatically a probability law or an invariant law.
- A formal generator display does not establish a closed operator domain.