Plain-English statement
- The derivative of the Euclidean radial cutoff transports to raw finite Pi space through `WithLp.toLp 2` by the chain rule. This is the cutoff-side `HasFDerivAt` producer consumed by the finite-box cutoff-smul route. It is pointwise and proves no support containment, integrability, tail limit, or integration-by-parts identity.
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Proof architecture
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Syntax used on this page
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Lean statement
theorem hasFDerivAt_radialSmoothCutoff_comp_toLp
{n : ℕ} {R : ℝ} (hR : 0 < R) (x : Fin (n + 1) → ℝ) :
HasFDerivAt
(fun z => Cutoff.radialSmoothCutoff R
(WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
((fderiv ℝ
(Cutoff.radialSmoothCutoff R :
EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
(WithLp.toLp 2 x)).comp
(PiLp.continuousLinearEquiv
2 ℝ (fun _ : Fin (n + 1) => ℝ)).symm.toContinuousLinearMap)
x := by
let e : EuclideanSpace ℝ (Fin (n + 1)) ≃L[ℝ] (Fin (n + 1) → ℝ) :=
PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin (n + 1) => ℝ)
have htoLp : HasFDerivAt
(fun z : Fin (n + 1) → ℝ =>
(WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1))))
e.symm.toContinuousLinearMap x := by
simpa [e] using
(PiLp.hasFDerivAt_toLp (𝕜 := ℝ)
(E := fun _ : Fin (n + 1) => ℝ) 2 x)
have hcutoff : HasFDerivAt
(Cutoff.radialSmoothCutoff R :
EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
(fderiv ℝ
(Cutoff.radialSmoothCutoff R :
EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
(WithLp.toLp 2 x))
(WithLp.toLp 2 x) :=
((Cutoff.radialSmoothCutoff_contDiff hR).differentiable
(WithTop.coe_ne_zero.mpr WithTop.top_ne_zero)
(WithLp.toLp 2 x)).hasFDerivAt
simpa [Function.comp_def, e] using hcutoff.comp x htoLp
/-- For an integrable finite Pi-space vector field, the `L¹` norm of the
radial-cutoff gradient applied to that field vanishes as the cutoff scale tends
to infinity.
The domination retains the operator norm of the inverse `PiLp` equivalence:
the raw Pi norm is not identified with the Euclidean `L²` norm. This theorem
only controls the cutoff-gradient cross term; it proves no source-field
integrability, main-term convergence, integration by parts, or invariant-law
statement. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:114published source at 7bcd37294df1
Proof architecture
Chewi Ch.1 cutoff-smul route: expose the Euclidean radial-cutoff derivative as a raw finite-Pi-space HasFDerivAt producer
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `simpa` closes the goal after a controlled simplification of a typed result.
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
- Measurability is represented explicitly or must be supplied by a dependency.
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Totalized `fderiv` values must not be read as a differentiability theorem.
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.
- 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.