signedFaceTermSum_eq_zero_of_boundary_component_eq_zero

Plain-English statement

- 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.

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 signedFaceTermSum_eq_zero_of_boundary_component_eq_zero
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (hupper : ∀ (i : Fin (n + 1)) (x : Fin n → ℝ),
      F (i.insertNth (b i) x) i = 0)
    (hlower : ∀ (i : Fin (n + 1)) (x : Fin n → ℝ),
      F (i.insertNth (a i) x) i = 0) :
    ∑ 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) = 0 := by
  simp [hupper, hlower]

/-- Version of `signedFaceTermSum_eq_zero_of_boundary_component_eq_zero`
with boundary values expressed by `Function.update`.

This is often the more convenient shape for later support or cutoff lemmas:
if replacing coordinate `i` by either endpoint forces the `i`-th component of
`F` to vanish, then the signed face-term sum vanishes.  This still assumes the
boundary values directly; compact-support and tail-decay proofs remain
separate obligations. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:689published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 boundary route: explicit zero normal components on all finite-box faces imply the signed face-term sum vanishes

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`simp` normalizes through registered definitional and theorem rewrites.

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