update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo

Plain-English statement

- 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.

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 update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (hoff : ∀ x ∉ (Set.univ.pi fun i => Set.Ioo (a i) (b i)), F x = 0) :
    (∀ (i : Fin (n + 1)) (x : Fin (n + 1) → ℝ),
        F (Function.update x i (b i)) i = 0) ∧
      (∀ (i : Fin (n + 1)) (x : Fin (n + 1) → ℝ),
        F (Function.update x i (a i)) i = 0) := by
  constructor
  · intro i x
    have hxnot : Function.update x i (b i) ∉
        (Set.univ.pi fun j => Set.Ioo (a j) (b j)) := by
      intro hx
      have hlt : b i < b i := by
        simpa using (hx i (Set.mem_univ _)).2
      exact (lt_irrefl (b i)) hlt
    exact congrArg (fun y => y i) (hoff (Function.update x i (b i)) hxnot)
  · intro i x
    have hxnot : Function.update x i (a i) ∉
        (Set.univ.pi fun j => Set.Ioo (a j) (b j)) := by
      intro hx
      have hlt : a i < a i := by
        simpa using (hx i (Set.mem_univ _)).1
      exact (lt_irrefl (a i)) hlt
    exact congrArg (fun y => y i) (hoff (Function.update x i (a i)) hxnot)

/-- Off-open-box vanishing implies Mathlib's finite-box signed face-term sum
is zero.

This composes `update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo`
with the update-shaped face-term producer.  It still does not prove how the
off-open-box vanishing hypothesis arises; compact support and tail decay remain
separate leaves. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:741published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 boundary route: off-open-box vanishing implies update-to-boundary normal components are zero

Lean proof walkthrough

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`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.

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