signedFaceTermSum_eq_zero_of_update_boundary_component_eq_zero

Plain-English statement

- 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.

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_update_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 + 1) → ℝ),
      F (Function.update x i (b i)) i = 0)
    (hlower : ∀ (i : Fin (n + 1)) (x : Fin (n + 1) → ℝ),
      F (Function.update x i (a i)) 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 :=
  signedFaceTermSum_eq_zero_of_boundary_component_eq_zero a b F
    (fun i x => by
      simpa [Function.update_eq_self] using hupper i (i.insertNth (b i) x))
    (fun i x => by
      simpa [Function.update_eq_self] using hlower i (i.insertNth (a i) x))

/-- If a Pi-space vector field vanishes outside the open box
`Set.univ.pi (fun i => Set.Ioo (a i) (b i))`, then its normal components
vanish after updating any coordinate to either endpoint.

This is a direct boundary producer for later compact-support or cutoff
arguments: those arguments can prove the off-open-box vanishing hypothesis,
and this leaf converts it into the update-boundary hypotheses used by the
finite-box face-term lemmas.  It does not prove compact support, tail decay,
whole-space limits, weighted integration by parts, generator domains, invariant
laws, or reversibility. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:712published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 boundary route: update-to-boundary zero component hypotheses imply the finite-box 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.
`simpa` closes the goal after a controlled simplification of a typed result.

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A Gibbs expression is not automatically a probability law or an invariant law.
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A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.