fderiv_norm_div_bound · Samplinglib

Plain-English statement

- Scaling the norm by a positive radius gives an operator-norm derivative bound of `1 / R`. Mathlib's totalized `fderiv` makes the statement valid at the origin as well.

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.

Lean learning studio · mathematics → formal proof

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Syntax used on this page

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Lean statement

theorem fderiv_norm_div_bound [NormedSpace ℝ E] {R : ℝ} (hR : 0 < R) (x : E) :
    ‖fderiv ℝ (fun y : E => ‖y‖ / R) x‖ ≤ 1 / R := by
  have hLip : LipschitzWith ⟨1 / R, by positivity⟩ (fun y : E => ‖y‖ / R) :=
    LipschitzWith.of_dist_le_mul fun y z => by
      have hnorm : |‖y‖ - ‖z‖| ≤ ‖y - z‖ := abs_norm_sub_norm_le y z
      simp only [Real.dist_eq]
      have hdiv : ‖y‖ / R - ‖z‖ / R = (‖y‖ - ‖z‖) / R := by ring
      rw [hdiv, abs_div, abs_of_pos hR]
      calc
        |‖y‖ - ‖z‖| / R ≤ ‖y - z‖ / R :=
          div_le_div_of_nonneg_right hnorm hR.le
        _ = 1 / R * ‖y - z‖ := by ring
        _ = 1 / R * dist y z := by rw [dist_eq_norm]
  exact norm_fderiv_le_of_lipschitz ℝ hLip

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

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:182published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 exhaustion base: the totalized derivative of x maps to norm x divided by R has operator norm at most 1/R

Lean proof walkthrough

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Why the statement has this shape

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Hidden assumptions and non-claims

Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.