ASTIS mathematical exposition

Off-open-box vanishing gives zero box divergence integral

AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo · theorem · Teaching coverage

Statement

Let a≤b and let F:P→P be continuous on K, have derivative A(x) at each x∈O∖s for a countable s, and have integrable trace τ_A on K. Assume additionally: F(x)=0 for every x∉O. Then the integral of δF over K is zero.

\[\int_K\delta F(x)\,dx=0.\]

All objects and hypotheses

n∈ℕ, d=n+1≥1, P=(Fin d→ℝ) with its usual supremum norm, V=EuclideanSpace ℝ (Fin d) with its ℓ² norm. T:P→V is WithLp.toLp 2, a continuous linear equivalence, and e=T⁻¹=WithLp.ofLp.
a,b∈P; K=[a,b]={x:∀i,a_i≤x_i≤b_i} is the closed box and O=∏_i(a_i,b_i) is the open box. All unspecified integrals and a.e. assertions use Lebesgue volume on P, restricted to K when indicated.
F:P→P and A:P→(P→L[ℝ]P) is a supplied linear-map field. Write δF(x)=Div(T∘F∘T⁻¹)(Tx) and τ_A(x)=Σ_i(A(x)e_i)_i.
a≤b coordinatewise (zero-width coordinates are allowed).
s⊆P is countable; F is continuous on K; for every x∈O∖s, HasFDerivAt F (A x) x.
τ_A is integrable on K with respect to volume.
For each i, K_hat_i=∏_{j≠i}[a_j,b_j] is parametrized by Fin n→ℝ; I_i^c inserts c in coordinate i, retaining the other coordinates in their original order. Φ_a,b(F) is the sum of upper-face integrals of F_i minus lower-face integrals of F_i.
F(x)=0 for every x∉O.

Notation and interpretation

Notation used below: For a raw field F:P→P, W_F=T∘F∘T⁻¹ and δF(x) is coordinateDivergence W_F at Tx. For a supplied linear-map field A, τ_A is its coordinate trace.
\[W_F:=T\circ F\circ T^{-1},\qquad\delta F(x):=\operatorname{Div}W_F(Tx),\qquad\tau_A(x):=\sum_i(A(x)e_i)_i.\]

Mathematical proof

1. Produce the zero signed face sum

Every vector with one coordinate equal to an endpoint is outside O. Thus the off-box zero premise kills all endpoint normal components and their face integrals.

\[F|_{P\setminus O}=0\Longrightarrow\Phi_{a,b}(F)=0.\]

Corresponding Lean step

signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo a b F with the stated boundary/support premises.

2. Combine with the conditional zero-face theorem

The original regularity and trace-integrability data satisfy the finite-box divergence theorem. Its zero-face corollary uses the just-proved Φ=0 to give the zero box integral.

\[\int_K\delta F\,dx=\Phi_{a,b}(F)=0.\]

Corresponding Lean step

integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable a b hle F F' s hs Hc Hd Hi_trace (...).

Lean statement · integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo

The derivative field F′ is supplied and is required to be correct only on O∖s. The chosen boundary/support premise replaces hfaces, but does not replace any regularity or integrability premise.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ) (hle : a ≤ b)
    (F : (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 F (Set.Icc a b))
    (Hd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
      HasFDerivAt F (F' x) x)
    (Hi_trace : IntegrableOn (fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i)
      (Set.Icc a b) volume)
    (hoff : ∀ x ∉ (Set.univ.pi fun i => Set.Ioo (a i) (b i)), F x = 0) :
    ∫ x in Set.Icc a b, coordinateDivergence
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
          (WithLp.toLp 2 (F (WithLp.ofLp y)) :
            EuclideanSpace ℝ (Fin (n + 1))))
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) = 0

Exact module and namespace context

Lean proof · integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo

The source first every vector with one coordinate equal to an endpoint is outside O. Thus the off-box zero premise kills all endpoint normal components and their face integrals. It finishes as follows: The original regularity and trace-integrability data satisfy the finite-box divergence theorem. Its zero-face corollary uses the just-proved Φ=0 to give the zero box integral. Intermediate steps below identify the actual helper calls and the conditions each one needs.

theorem integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ) (hle : a ≤ b)
    (F : (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 F (Set.Icc a b))
    (Hd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
      HasFDerivAt F (F' x) x)
    (Hi_trace : IntegrableOn (fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i)
      (Set.Icc a b) volume)
    (hoff : ∀ x ∉ (Set.univ.pi fun i => Set.Ioo (a i) (b i)), F x = 0) :
    ∫ x in Set.Icc a b, coordinateDivergence
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
          (WithLp.toLp 2 (F (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_integrableOn_trace_of_hasFDerivAt_off_countable
    a b hle F F' s hs Hc Hd Hi_trace
    (signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo a b F hoff)

/-- Finite-box coordinate-divergence integral vanishes when the vector field's
support is contained in the open Pi-box.

This composes the support-to-face producer with the finite-box divergence
wrapper.  It is not a compact-support theorem, a cutoff construction, a
whole-space integration-by-parts theorem, or an invariant-law result. -/

Exact module and namespace context

Scope and omitted-condition boundaries

Only the finite box K is integrated. No passage to whole space, weighted integration by parts, generator domain, stationary law, or invariance follows from this wrapper alone.
Derivative values at the boundary and on the countable exceptional set are not prescribed; equality of integrands is used a.e.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

No direct Mathlib call recorded; see the ASTIS parents.

Mathematical sources

Current ASTIS source — Exact statement and actual proof/construction authority; raw code intentionally omitted from this packet.
Existing curated module card — Existing declaration-specific attribution entry, read as documentation without a new source-equivalence verdict.
Called library/source declaration: AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable — Exact ASTIS parent called in the proof steps above; this anchor adds no source-equivalence verdict.
Called library/source declaration: AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo — Exact ASTIS parent called in the proof steps above; this anchor adds no source-equivalence verdict.
Existing focused test — Exact named declaration invocation located in an existing example; no test was run.

ASTIS prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.