support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo

Plain-English statement

- Topological-support containment implies plain function-support containment inside a finite Pi-open box. This is the bridge needed by the finite-box cutoff route: Mathlib's smooth cutoff theorem naturally returns `tsupport`, while the already-compiled zero-face handoffs are phrased using `Function.support`. The lemma is only a support-API conversion; it does not construct a cutoff or prove any derivative, tail, or whole-space integration-by-parts statement.

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 support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo
    {n : ℕ} {a b : Fin (n + 1) → ℝ}
    {χ : (Fin (n + 1) → ℝ) → ℝ}
    (hχ : tsupport χ ⊆ Set.univ.pi (fun i => Set.Ioo (a i) (b i))) :
    Function.support χ ⊆ Set.univ.pi (fun i => Set.Ioo (a i) (b i)) := by
  exact (subset_tsupport χ).trans hχ

/-- Smooth finite-dimensional cutoff localized inside a Pi-open box, with both
topological-support and plain function-support conclusions.

This packages `exists_contDiff_cutoff_tsupport_subset_univ_pi_Ioo` with
`support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo`, so downstream
finite-box support lemmas can consume the cutoff directly.  It remains local:
no exhausting cutoff family, derivative bookkeeping, tail limit, weighted IBP,
generator-domain theorem, invariant law, or reversibility is asserted. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:835published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff route: convert Mathlib topological-support containment for a scalar cutoff into the plain Function.support containment consumed by finite-box zero-face handoffs

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Hidden assumptions and non-claims

Pointwise support, topological support, and compact support retain distinct meanings.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.