exists_contDiff_eq_one_tsupport_subset

Plain-English statement

- A compact subset of an open set admits a smooth compactly supported plateau in that set. The function takes values in `[0, 1]` and is identically one on the compact set.

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Lean statement

theorem exists_contDiff_eq_one_tsupport_subset
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
    ∃ χ : E → ℝ,
      Function.support χ ⊆ U ∧ tsupport χ ⊆ U ∧ HasCompactSupport χ ∧
        ContDiff ℝ (⊤ : ℕ∞) χ ∧ Set.range χ ⊆ Set.Icc 0 1 ∧ Set.EqOn χ 1 K := by
  obtain ⟨L, hL, hK_intL, hL_U⟩ := exists_compact_between hK hU hKU
  obtain ⟨g, hg_support, hg_contDiff, hg_range⟩ :=
    (isOpen_interior : IsOpen (interior L)).exists_contDiff_support_eq
      (E := E) (n := (⊤ : ℕ∞))
  have hg_pos : ∀ x ∈ K, 0 < g x := by
    intro x hx
    have hx_support : x ∈ Function.support g := by
      rw [hg_support]
      exact hK_intL hx
    exact lt_of_le_of_ne
      (hg_range (Set.mem_range_self x)).1 (Ne.symm hx_support)
  obtain ⟨m, hm_pos, hm_le⟩ :=
    hK.exists_forall_le' hg_contDiff.continuous.continuousOn hg_pos
  have hm_half_pos : 0 < m / 2 := half_pos hm_pos
  let χ : E → ℝ := fun x =>
    Real.smoothTransition ((g x - m / 2) / (m / 2))
  have hχ_support_superlevel :
      Function.support χ ⊆ {x : E | m / 2 ≤ g x} := by
    intro x hx
    change χ x ≠ 0 at hx
    have harg : 0 < (g x - m / 2) / (m / 2) := by
      apply lt_of_not_ge
      intro hnonpos
      apply hx
      exact Real.smoothTransition.zero_of_nonpos hnonpos
    have hnum : 0 < g x - m / 2 :=
      (div_pos_iff_of_pos_right hm_half_pos).mp harg
    exact (sub_pos.mp hnum).le
  have hsuperlevel_closed : IsClosed {x : E | m / 2 ≤ g x} :=
    isClosed_le continuous_const hg_contDiff.continuous
  have hχ_tsupport_superlevel :
      tsupport χ ⊆ {x : E | m / 2 ≤ g x} := by
    exact closure_minimal hχ_support_superlevel hsuperlevel_closed
  have hsuperlevel_intL : {x : E | m / 2 ≤ g x} ⊆ interior L := by
    intro x hx
    have hgx_pos : 0 < g x := hm_half_pos.trans_le hx
    have hx_support : x ∈ Function.support g := hgx_pos.ne'
    rwa [hg_support] at hx_support
  have hχ_tsupport_L : tsupport χ ⊆ L :=
    hχ_tsupport_superlevel.trans (hsuperlevel_intL.trans interior_subset)
  have hχ_tsupport_U : tsupport χ ⊆ U := hχ_tsupport_L.trans hL_U
  have hχ_support_U : Function.support χ ⊆ U :=
    (subset_tsupport χ).trans hχ_tsupport_U
  have hχ_compact : HasCompactSupport χ := by
    rw [hasCompactSupport_def]
    exact hL.of_isClosed_subset isClosed_closure hχ_tsupport_L
  have hχ_contDiff : ContDiff ℝ (⊤ : ℕ∞) χ := by
    apply Real.smoothTransition.contDiff.comp
    exact (hg_contDiff.sub contDiff_const).div_const (m / 2)
  have hχ_range : Set.range χ ⊆ Set.Icc (0 : ℝ) 1 := by
    rintro _ ⟨x, rfl⟩
    exact ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
  have hχ_one : Set.EqOn χ 1 K := by
    intro x hx
    apply Real.smoothTransition.one_of_one_le
    rw [one_le_div hm_half_pos]
    linarith [hm_le x hx]
  exact ⟨χ, hχ_support_U, hχ_tsupport_U, hχ_compact,
    hχ_contDiff, hχ_range, hχ_one⟩

end Plateau

end Cutoff
end Calculus
end Analysis
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:408published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff root: a compact set inside an open finite-dimensional neighborhood admits a smooth [0,1] plateau with compact support inside that neighborhood

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
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`rw` rewrites by an established identity.
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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

Pointwise support, topological support, and compact support retain distinct meanings.
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.