ControlApproximationEffect.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 981 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned Carleson blueprint

9 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

lemma rcarleson'_restrict line 24 source
depends on sorry what this means
Statement uses: NNReal, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Nat, Real, Complex, Function.Periodic, HMul.hMul, Real.pi, MeasureTheory.MemLp, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Ioc, LE.le, ENNReal, MeasureTheory.eLpNorm, carlesonOperatorReal, K and 1 more
Proof uses: MeasureTheory.AEStronglyMeasurable, Function.Periodic.aestronglyMeasurable, instFactLtRealOfNatHMulPi, And.left, HAdd.hAdd, LT.lt, Top.top, Eq.mp, congrArg, Eq.symm, zero_add, Set.indicator, Neg.neg, Trans.trans, Eq, rfl, LE.le.trans', MeasureTheory.eLpNorm_add_le, Eq.mpr, id and 128 more
210 instances, coercions and auxiliary constants left out.

Tactic trace: 60 steps
Open to load the checked before-and-after goal states.
def C_distribution_carlesonOperatorReal_le The constant used in C_distribution_carlesonOperatorReal_le line 87 source
Source documentation
The constant used in C_distribution_carlesonOperatorReal_le.

depends on sorry what this means
Statement uses: NNReal
Proof uses: HMul.hMul, Inv.inv, OfNat.ofNat, C_carleson_hasStrongType, Nat, MeasureTheory.C_distribution_le_of_eLpNorm_le
12 instances, coercions and auxiliary constants left out.
```
def C_distribution_carlesonOperatorReal_le (δ ε p : ℝ≥0) : ℝ≥0 :=
…
```
lemma C_distribution_carlesonOperatorReal_le_pos line 90 source
depends on sorry what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, C_distribution_carlesonOperatorReal_le
Proof uses: id, mul_pos, Inv.inv, HMul.hMul, C_carleson_hasStrongType, Nat, MeasureTheory.C_distribution_le_of_eLpNorm_le, of_eq_true, Eq.trans, True, congrArg, mul_inv_rev, MeasureTheory.C_distribution_le_of_eLpNorm_le_pos
56 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma distribution_carlesonOperatorReal_le line 96 source
depends on sorry what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Membership.mem, Set, Set.Ioo, Nat, Real, Complex, Function.Periodic, HMul.hMul, Real.pi, MeasureTheory.AEStronglyMeasurable, MeasureTheory.MeasureSpace.volume, LE.le, ENNReal, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, Set.Ioc, C_distribution_carlesonOperatorReal_le and 3 more
Proof uses: MeasureTheory.distribution_le_of_eLpNorm_le, LT.lt.trans, zero_lt_one, And.left, Measurable.aestronglyMeasurable, carlesonOperatorReal_measurable, HAdd.hAdd, LE.le.trans', Set.Ioo_subset_Ioc_self, Set.Ioo_subset_Ioo_right, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance and 90 more
199 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
def C_control_approximation_effect The constant used in C_control_approximation_effect line 127 source
Source documentation
The constant used in C_control_approximation_effect.

depends on sorry what this means
Statement uses: NNReal
Proof uses: Min.min, HMul.hMul, OfNat.ofNat, HDiv.hDiv, Inv.inv, HPow.hPow, Real, Real.toNNReal, Real.pi, HSub.hSub, C_distribution_carlesonOperatorReal_le
27 instances, coercions and auxiliary constants left out.
```
def C_control_approximation_effect (δ ε p : ℝ≥0) : ℝ≥0 :=
…
```
lemma C_control_approximation_effect_pos line 131 source
depends on sorry what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, C_control_approximation_effect
Proof uses: lt_min, HMul.hMul, HDiv.hDiv, Inv.inv, HPow.hPow, Real, Real.toNNReal, Real.pi, HSub.hSub, C_distribution_carlesonOperatorReal_le, mul_pos, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, Nat, div_pos, inv_pos_of_pos and 8 more
64 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma C_control_approximation_effect_property line 136 source
depends on sorry what this means
Statement uses: NNReal, LE.le, OfNat.ofNat, ENNReal, HMul.hMul, C_control_approximation_effect, HPow.hPow, Real, Nat, ENNReal.ofReal, Real.pi, HSub.hSub, Inv.inv, HDiv.hDiv
Proof uses: Trans.trans, Eq, Real.toNNReal, rfl, mul_le_mul', Eq.mpr, id, Eq.trans, congrArg, congrFun', Eq.symm, Nat.cast_one, congr, min_le_left, C_distribution_carlesonOperatorReal_le, le_refl, ENNReal.coe_inv, of_eq_true, Not, True and 38 more
139 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma C_control_approximation_effect_le line 150 source
depends on sorry what this means
Statement uses: NNReal, LE.le, ENNReal, C_control_approximation_effect, C_distribution_carlesonOperatorReal_le, Real.toNNReal, HDiv.hDiv, Real, HMul.hMul, OfNat.ofNat, Nat, Real.pi
Proof uses: Eq.mpr, id, Eq, min_le_right, Inv.inv, HPow.hPow, HSub.hSub
54 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma control_approximation_effect This is Lemma 11.6.4 (partial Fourier sums of small) in the blueprint line 157 source
Source comment
This is Lemma 11.6.4 (partial Fourier sums of small) in the blueprint.

depends on sorry what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, MeasureTheory.AEStronglyMeasurable, MeasureTheory.MeasureSpace.volume, Function.Periodic, HMul.hMul, Nat, Real.pi, Membership.mem, Set, Set.Ioo, LE.le, ENNReal, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, Set.Ioc, C_control_approximation_effect and 4 more
Proof uses: Trans.trans, Eq, HAdd.hAdd, HDiv.hDiv, carlesonOperatorReal, K, Function.comp, RingHom, starRingEnd, ENNReal.ofReal, operatorBound, rfl, MeasureTheory.distribution_mono_left, Eq.mpr, Filter.Eventually, MeasureTheory.ae, MeasureTheory.Measure, id, congrArg, propext and 110 more
260 instances, coercions and auxiliary constants left out.

Tactic trace: 75 steps
Open to load the checked before-and-after goal states.
```
lemma control_approximation_effect {δ ε : ℝ≥0} (δpos : 0 < δ)
  {g : ℝ → ℂ} (g_measurable : AEStronglyMeasurable g)
  (g_periodic : g.Periodic (2 * π))
  {p : ℝ≥0} (hp : p ∈ Set.Ioo 1 2)
  (g_bound : eLpNorm g p (volume.restrict (Set.Ioc 0 (2 * π))) ≤ C_control_approximation_effect δ ε p) :
    distribution (fun x ↦ ⨆ N, ‖S_ N g x‖ₑ) δ (volume.restrict (Set.Ioc 0 (2 * π))) ≤ ε := by
…
```
Mentions: g, C_control_approximation_effect