Plain-English statement
- Box-level signed-face divergence theorem wrapper for ASTIS coordinate divergence, using Mathlib's trace-integrability hypothesis directly. Compared with `integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable`, this version no longer asks callers to provide the `hdiv_ae` representation bridge or coordinate-divergence integrability. Both are derived from the open-box/off-countable derivative hypothesis and the explicit trace-integrability assumption. It still proves only the finite-box signed face-term formula. It does not prove trace integrability for a concrete vector field, whole-space/no-boundary limits, weighted IBP, generator domains, invariant Gibbs law, reversibility, stationarity, or KL/FI dissipation.
Read the mathematics first, then descend into Lean
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Proof architecture
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Lean statement
theorem integral_coordinateDivergence_toPi_box_of_integrableOn_trace_of_hasFDerivAt_off_countable
{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) :
∫ 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))) =
∑ i : Fin (n + 1),
((∫ x in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove),
F (i.insertNth (b i) x) i) -
∫ x in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove),
F (i.insertNth (a i) x) i) := by
have hdiv_ae := coordinateDivergence_wrapped_toPi_trace_ae_of_hasFDerivAt_off_countable
a b F F' s hs Hd
have Hi_coord :=
integrableOn_coordinateDivergence_wrapped_of_integrableOn_trace_of_hasFDerivAt_off_countable
a b F F' s hs Hd Hi_trace
exact integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable
a b hle F F' s hs Hc Hd hdiv_ae Hi_coord
/-- If the normal component of a Pi-space vector field vanishes on every
lower and upper face of a finite box, then Mathlib's signed face-term sum is
zero.
This is a boundary-value producer for the finite-box divergence route. It only
turns explicit componentwise zero boundary values into a zero signed face term;
it does not prove compact support, tail decay, weighted integration by parts,
whole-space limits, generator domains, invariant laws, or reversibility. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:651published source at 7bcd37294df1
Proof architecture
Chewi Ch.1 calculus root: finite-box signed face-term formula for ASTIS coordinateDivergence with only Mathlib trace-integrability as the integrability input
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.
- `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.
- Pointwise support, topological support, and compact support retain distinct meanings.
- Almost-everywhere hypotheses depend on the stated measure and representative.
- A Gibbs expression is not automatically a probability law or an invariant law.
- A formal generator display does not establish a closed operator domain.