Plain-English statement
- Canonical-`fderiv` scalar-support version of the cutoff-smul finite-box zero integral handoff. This removes only the supplied derivative-field parameter `G'`, replacing it by `fderiv ℝ G` under open-box differentiability of `G`. It remains a finite-box handoff and does not construct a cutoff, prove trace integrability, pass to whole space, prove weighted integration by parts, or prove an invariant law.
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Proof architecture
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Lean statement
theorem integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_fderiv
{n : ℕ}
(a b : Fin (n + 1) → ℝ) (hle : a ≤ b)
(χ : (Fin (n + 1) → ℝ) → ℝ)
(χ' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] ℝ)
(G : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
(s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable)
(hχc : ContinuousOn χ (Set.Icc a b))
(hGc : ContinuousOn G (Set.Icc a b))
(hχd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
HasFDerivAt χ (χ' x) x)
(hGd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
DifferentiableAt ℝ G x)
(Hi_trace : IntegrableOn
(fun x => ∑ i, ((χ x • fderiv ℝ G x + (χ' x).smulRight (G x))
(Pi.single i (1 : ℝ))) i)
(Set.Icc a b) volume)
(hχsupp : Function.support χ ⊆
(Set.univ.pi fun i => Set.Ioo (a i) (b i))) :
∫ x in Set.Icc a b, coordinateDivergence
(fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
(WithLp.toLp 2 ((χ (WithLp.ofLp y)) • G (WithLp.ofLp y)) :
EuclideanSpace ℝ (Fin (n + 1))))
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) = 0 := by
exact integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_regularity
a b hle χ χ' G (fun x => fderiv ℝ G x) s hs hχc hGc hχd
(fun x hx => (hGd x hx).hasFDerivAt) Hi_trace hχsupp
/-- Scalar-support version of
`integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous`.
It uses closed-box continuity of the product-rule trace to discharge the
compact-box trace-integrability side condition, but remains only a finite-box
cutoff-smul handoff. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1679published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 finite-box cutoff route: replace the supplied vector-field derivative parameter by the canonical `fderiv ℝ G` in the scalar-support zero integral handoff
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
- 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.
- Pointwise support, topological support, and compact support retain distinct meanings.
- A Gibbs expression is not automatically a probability law or an invariant law.
- A formal generator display does not establish a closed operator domain.
- A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.