finiteCoord_weightedDivergence_langevinGenerator_algebra

Plain-English statement

- Finite-coordinate aggregation of supplied product-rule and chain-rule identities for the weighted-divergence form of the Langevin generator. Here `divCoord i` represents the already-supplied coordinate derivative `∂ᵢ (rho * ∂ᵢ f)`, `hessDiag i` represents `∂ᵢᵢ f`, and `gradRho`, `gradV`, `gradF` are coordinate representatives. This theorem only sums those coordinate algebra facts and factors the scalar `rho`; it does not define or prove partial derivatives, gradients, divergence, or the Laplacian.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem finiteCoord_weightedDivergence_langevinGenerator_algebra
    {ι : Type*} [Fintype ι]
    {rho : ℝ} {divCoord hessDiag gradRho gradV gradF : ι → ℝ}
    (hdiv : ∀ i, divCoord i = rho * hessDiag i + gradRho i * gradF i)
    (hgrad : ∀ i, gradRho i = -rho * gradV i) :
    (∑ i, divCoord i) =
      rho * ((∑ i, hessDiag i) - ∑ i, gradV i * gradF i) := by
  have hsumHess : (∑ i, rho * hessDiag i) = rho * ∑ i, hessDiag i := by
    rw [Finset.mul_sum]
  have hsumDrift : (∑ i, (-rho * gradV i) * gradF i) =
      -rho * ∑ i, gradV i * gradF i := by
    calc
      (∑ i, (-rho * gradV i) * gradF i) =
          ∑ i, -rho * (gradV i * gradF i) := by
        exact Finset.sum_congr rfl (fun i _ => by ring)
      _ = -rho * ∑ i, gradV i * gradF i := by
        rw [Finset.mul_sum]
  calc
    (∑ i, divCoord i) =
        ∑ i, (rho * hessDiag i + gradRho i * gradF i) := by
      exact Finset.sum_congr rfl (fun i _ => hdiv i)
    _ = ∑ i, (rho * hessDiag i + (-rho * gradV i) * gradF i) := by
      exact Finset.sum_congr rfl (fun i _ => by rw [hgrad i])
    _ = rho * ((∑ i, hessDiag i) - ∑ i, gradV i * gradF i) := by
      rw [Finset.sum_add_distrib, hsumHess, hsumDrift]
      ring

/-- Named finite-coordinate wrapper for
`finiteCoord_weightedDivergence_langevinGenerator_algebra`.

The hypotheses `hlap` and `hinner` are supplied identifications of the
coordinate sums with a named Laplacian scalar and a named gradient inner-product
scalar.  The theorem does not prove those identifications. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:114published source at 7bcd37294df1

Proof architecture

Chewi SDE/DENS root: aggregate coordinate identities `∂ᵢ(rho ∂ᵢf)` into the finite-sum Langevin divergence-form expression

Lean proof walkthrough

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Hidden assumptions and non-claims

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