Plain-English statement
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector field when the scalar cutoff's topological support is contained in the open Pi-box. This is a `tsupport`-API variant of the scalar-support finite-box handoff. It still assumes the cutoff-smul field's closed-box continuity, off-countable Frechet differentiability, and trace integrability; it does not construct a cutoff family, prove tail decay, or derive whole-space weighted IBP.
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Proof architecture
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Syntax used on this page
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Lean statement
theorem integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_tsupport_subset_univ_pi_Ioo
{n : ℕ}
(a b : Fin (n + 1) → ℝ) (hle : a ≤ b)
(χ : (Fin (n + 1) → ℝ) → ℝ)
(G : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
(F' : (Fin (n + 1) → ℝ) →
(Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
(s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable)
(Hc : ContinuousOn (fun x => χ x • G x) (Set.Icc a b))
(Hd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
HasFDerivAt (fun x => χ x • G x) (F' x) x)
(Hi_trace : IntegrableOn (fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i)
(Set.Icc a b) volume)
(hχtsupp : tsupport 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 :=
integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo
a b hle χ G F' s hs Hc Hd Hi_trace
(support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo hχtsupp)
/-- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector
field, deriving the continuity and off-countable Frechet differentiability
hypotheses from separate cutoff and vector-field regularity assumptions.
The trace integrability of the product-rule derivative remains an explicit
assumption. This theorem is still finite-box only: it does not prove smooth
cutoff construction, tail limits, whole-space weighted IBP, generator domains,
invariant laws, reversibility, or KL/FI. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1492published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 finite-box cutoff route: scalar cutoff topological support inside the open Pi-box implies the cutoff-smul coordinate-divergence box integral is zero under the existing trace/differentiability assumptions
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
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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 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.