radialSmoothCutoff_contDiff

Plain-English statement

- For positive scale, the radial cutoff is infinitely differentiable.

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 radialSmoothCutoff_contDiff [InnerProductSpace ℝ E] {R : ℝ} (hR : 0 < R) :
    ContDiff ℝ (⊤ : ℕ∞) (radialSmoothCutoff R : E → ℝ) := by
  rw [contDiff_iff_contDiffAt]
  intro x
  by_cases hx : ‖x‖ < R
  · have heq : ∀ᶠ y in 𝓝 x, radialSmoothCutoff R y = 1 := by
      have hball : Metric.ball x (R - ‖x‖) ∈ 𝓝 x :=
        Metric.ball_mem_nhds x (sub_pos.mpr hx)
      filter_upwards [hball] with y hy
      apply radialSmoothCutoff_eq_one_of_norm_le hR
      rw [Metric.mem_ball, dist_eq_norm] at hy
      have hynorm : ‖y‖ ≤ ‖x‖ + ‖y - x‖ := by
        calc
          ‖y‖ = ‖x + (y - x)‖ := by congr 1; abel
          _ ≤ ‖x‖ + ‖y - x‖ := norm_add_le x (y - x)
      linarith
    have hconst : ContDiffAt ℝ (⊤ : ℕ∞) (fun _ : E => (1 : ℝ)) x :=
      contDiffAt_const
    apply hconst.congr_of_eventuallyEq
    filter_upwards [heq] with y hy
    simp [hy]
  · push Not at hx
    have hx_ne : x ≠ 0 := by
      intro hzero
      rw [hzero, norm_zero] at hx
      linarith
    have hnorm : ContDiffAt ℝ (⊤ : ℕ∞) (fun y : E => ‖y‖) x :=
      contDiffAt_norm ℝ hx_ne
    have hdiv : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ => t / R) :=
      contDiff_id.div_const R
    have hscaled : ContDiffAt ℝ (⊤ : ℕ∞) (fun y : E => ‖y‖ / R) x :=
      hdiv.contDiffAt.comp x hnorm
    exact smoothUnitCutoff_contDiff.contDiffAt.comp x hscaled

/-- A single positive constant controls the first derivative of every
positive-scale radial cutoff by `C / R`.  The quantifier order records the
scale-uniformity needed by cutoff exhaustion arguments. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:198published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 exhaustion base: smoothness of the positive-scale radial cutoff on real inner-product spaces

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Hidden assumptions and non-claims

A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.