Plain-English statement
- 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.
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Lean statement
theorem radialSmoothCutoff_fderiv_bound [InnerProductSpace ℝ E] :
∃ C : ℝ, 0 < C ∧ ∀ R : ℝ, 0 < R → ∀ x : E,
‖fderiv ℝ (radialSmoothCutoff R : E → ℝ) x‖ ≤ C / R := by
obtain ⟨C, hC_pos, hC_bound⟩ := smoothUnitCutoff_deriv_bounded
refine ⟨C, hC_pos, ?_⟩
intro R hR x
by_cases hxR : ‖x‖ < R
· have h_eq : ∀ᶠ y in 𝓝 x, radialSmoothCutoff R y = 1 := by
have hradius : 0 < R - ‖x‖ := sub_pos.mpr hxR
refine Metric.eventually_nhds_iff.mpr ⟨R - ‖x‖, hradius, ?_⟩
intro y hy
apply radialSmoothCutoff_eq_one_of_norm_le hR
rw [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 hfderiv_eq : fderiv ℝ (radialSmoothCutoff R : E → ℝ) x = 0 := by
have hconst : fderiv ℝ (fun _ : E => (1 : ℝ)) x = 0 := by simp
exact (Filter.EventuallyEq.fderiv_eq h_eq).trans hconst
rw [hfderiv_eq, norm_zero]
exact div_nonneg hC_pos.le hR.le
· push Not at hxR
have hx_ne : x ≠ 0 := by
intro hzero
rw [hzero, norm_zero] at hxR
linarith
have hnorm_diff : DifferentiableAt ℝ (fun y : E => ‖y‖ / R) x := by
have hnorm : DifferentiableAt ℝ (fun y : E => ‖y‖) x :=
(contDiffAt_norm ℝ hx_ne).differentiableAt WithTop.top_ne_zero
simpa only [div_eq_mul_inv] using hnorm.mul_const R⁻¹
have hcutoff_diff : DifferentiableAt ℝ smoothUnitCutoff (‖x‖ / R) :=
smoothUnitCutoff_contDiff.differentiable
(WithTop.coe_ne_zero.mpr WithTop.top_ne_zero) (‖x‖ / R)
have hchain :
fderiv ℝ (radialSmoothCutoff R : E → ℝ) x =
fderiv ℝ smoothUnitCutoff (‖x‖ / R) ∘L
fderiv ℝ (fun y : E => ‖y‖ / R) x := by
unfold radialSmoothCutoff
exact fderiv_comp x hcutoff_diff hnorm_diff
rw [hchain]
calc
‖fderiv ℝ smoothUnitCutoff (‖x‖ / R) ∘L
fderiv ℝ (fun y : E => ‖y‖ / R) x‖
≤ ‖fderiv ℝ smoothUnitCutoff (‖x‖ / R)‖ *
‖fderiv ℝ (fun y : E => ‖y‖ / R) x‖ :=
ContinuousLinearMap.opNorm_comp_le _ _
_ ≤ C * (1 / R) := by
apply mul_le_mul
· rw [← norm_deriv_eq_norm_fderiv]
exact hC_bound _
· exact fderiv_norm_div_bound hR x
· exact norm_nonneg _
· exact hC_pos.le
_ = C / R := by ring
/-- The totalized derivative of the radial cutoff vanishes throughout the
outer zero region, including its boundary sphere. At the boundary the cutoff
is a global minimum rather than locally constant; `IsLocalMin.fderiv_eq_zero`
records that distinction. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:235published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 exhaustion base: one constant controls the first derivative of every positive-scale radial cutoff by C/R
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.
- `have` creates a named intermediate mathematical fact.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
- `simp` normalizes through registered definitional and theorem rewrites.
- `apply` reduces the goal to the hypotheses of a reusable theorem.
- `refine` instantiates a reusable theorem while leaving explicit subgoals.
Why the statement has this shape
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
- 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.