Plain-English statement
- Clipping converges to the original coefficient in `L2`.
Read the mathematics first, then descend into Lean
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Proof architecture
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Syntax used on this page
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Lean statement
theorem tendsto_clipNat_toLp
{Ω : Type*} {m : MeasurableSpace Ω} {mu : Measure Ω}
{f : Ω → ℝ} (hf : MemLp f 2 mu) :
Tendsto
(fun n => (clipNat_memLp hf n).toLp (fun omega => clipNat n (f omega)))
atTop (𝓝 (hf.toLp f)) := by
let errorSq : ℕ → Ω → ℝ := fun n omega => (clipNat n (f omega) - f omega) ^ 2
have hmeas : ∀ n, AEStronglyMeasurable (errorSq n) mu := fun n =>
(((aestronglyMeasurable_clipNat hf.1 n).sub hf.1).pow 2)
have hboundInt : Integrable (fun omega => 4 * f omega ^ 2) mu :=
hf.integrable_sq.const_mul 4
have hbound : ∀ n, ∀ᵐ omega ∂mu, ‖errorSq n omega‖ ≤ 4 * f omega ^ 2 := by
intro n
filter_upwards [] with omega
have habs := abs_clipNat_sub_le n (f omega)
have hsq : |clipNat n (f omega) - f omega| ^ 2 ≤ (2 * |f omega|) ^ 2 :=
(sq_le_sq₀ (abs_nonneg _) (mul_nonneg (by norm_num) (abs_nonneg _))).2 habs
calc
‖errorSq n omega‖ = |clipNat n (f omega) - f omega| ^ 2 := by
simp [errorSq, Real.norm_eq_abs]
_ ≤ (2 * |f omega|) ^ 2 := hsq
_ = 4 * |f omega| ^ 2 := by ring
_ = 4 * f omega ^ 2 := by rw [sq_abs]
have hpoint : ∀ᵐ omega ∂mu,
Tendsto (fun n => errorSq n omega) atTop (𝓝 0) := by
filter_upwards [] with omega
have hconst : Tendsto (fun _ : ℕ => f omega) atTop (𝓝 (f omega)) :=
tendsto_const_nhds
simpa [errorSq] using ((tendsto_clipNat (f omega)).sub hconst).pow 2
have hintegral : Tendsto (fun n => ∫ omega, errorSq n omega ∂mu) atTop (𝓝 0) := by
simpa using tendsto_integral_of_dominated_convergence
(fun omega => 4 * f omega ^ 2) hmeas hboundInt hbound hpoint
have hnormSq : ∀ n,
‖(clipNat_memLp hf n).toLp (fun omega => clipNat n (f omega)) - hf.toLp f‖ ^ 2 =
∫ omega, errorSq n omega ∂mu := by
intro n
rw [← MemLp.toLp_sub]
exact norm_sq_toLp_eq_integral_sq ((clipNat_memLp hf n).sub hf)
apply tendsto_iff_norm_sub_tendsto_zero.mpr
have hsqrt : Tendsto
(fun n => Real.sqrt (∫ omega, errorSq n omega ∂mu)) atTop (𝓝 0) := by
simpa only [Function.comp_def, Real.sqrt_zero] using
(Real.continuous_sqrt.tendsto 0 |>.comp hintegral)
refine (tendsto_congr' ?_).mpr hsqrt
filter_upwards [] with n
rw [← hnormSq n, Real.sqrt_sq (norm_nonneg _)]
end CoefficientTruncation
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:105published source at 7bcd37294df1
Proof architecture
replace square-integrable real coefficients by bounded measurable coefficients without changing their L2 limit
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
- 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.
- Almost-everywhere hypotheses depend on the stated measure and representative.
- Density statements retain normalization and absolute-continuity prerequisites.