lineDeriv_expNegPotential_mul_fderiv_coordinate_eq

Plain-English statement

- Coordinate product rule for the explicit Gibbs weight multiplied by the coordinate derivative represented as `fderiv ℝ f y eᵢ`. Compared with `lineDeriv_expNegPotential_mul_eq_of_differentiableAt`, this also discharges the supplied derivative of `g` when `g y = fderiv ℝ f y eᵢ` and the total `fderiv` map of `f` is differentiable at `x`. It still does not replace `fderiv ℝ f y eᵢ` by `(gradient f y) i`, define or sum divergence, prove IBP, or prove invariant Gibbs law.

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 lineDeriv_expNegPotential_mul_fderiv_coordinate_eq
    {ι : Type*} [Fintype ι] [DecidableEq ι]
    {V f : EuclideanSpace ℝ ι → ℝ} {x : EuclideanSpace ℝ ι} (i : ι)
    (hV : DifferentiableAt ℝ V x)
    (hf : DifferentiableAt ℝ
      (fun y : EuclideanSpace ℝ ι => fderiv ℝ f y) x) :
    lineDeriv ℝ
        (fun y : EuclideanSpace ℝ ι =>
          Real.exp (-V y) *
            fderiv ℝ f y
              (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι))
        x
        (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι) =
      Real.exp (-V x) *
          iteratedFDeriv ℝ 2 f x
            ![(WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι),
              (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι)] -
        Real.exp (-V x) * (gradient V x) i *
          fderiv ℝ f x
            (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι) := by
  have hg :
      HasLineDerivAt ℝ
        (fun y : EuclideanSpace ℝ ι =>
          fderiv ℝ f y
            (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι))
        (iteratedFDeriv ℝ 2 f x
          ![(WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι),
            (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι)])
        x
        (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι) := by
    have hraw :=
      hasLineDerivAt_fderiv_apply_const_of_hasFDerivAt_fderiv
        (f := f) (x := x)
        (w := (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι))
        (v := (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι))
        (A := fderiv ℝ
          (fun y : EuclideanSpace ℝ ι => fderiv ℝ f y) x)
        hf.hasFDerivAt
    let direction : EuclideanSpace ℝ ι :=
      WithLp.toLp 2 (Pi.single i (1 : ℝ))
    have hvalue :
        (fderiv ℝ (fun y : EuclideanSpace ℝ ι => fderiv ℝ f y) x)
            direction direction =
          iteratedFDeriv ℝ 2 f x ![direction, direction] := by
      rw [iteratedFDeriv_two_apply]
      simp
    rw [← hvalue]
    exact hraw
  exact lineDeriv_expNegPotential_mul_eq_of_differentiableAt
    (V := V)
    (g := fun y : EuclideanSpace ℝ ι =>
      fderiv ℝ f y
        (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι))
    (x := x)
    (g' := iteratedFDeriv ℝ 2 f x
      ![(WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι),
        (WithLp.toLp 2 (Pi.single i (1 : ℝ)) : EuclideanSpace ℝ ι)])
    i hV hg

end LineDeriv
end Calculus
end Analysis
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:218published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Langevin root: compute the coordinate line derivative of `exp(-V) * fderiv f eᵢ`, producing the Gibbs-weighted diagonal iterated-derivative term and the potential-gradient product term

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.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`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.