ControlApproximationEffectContinuous.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

13 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

lemma rcarleson_exceptional_set_estimate line 22 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, Measurable, Set, MeasurableSet, LE.le, Norm.norm, HMul.hMul, Set.indicator, Membership.mem, ENNReal, carlesonOperatorReal, K, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, C10_0_1, Nat and 2 more
Proof uses: Trans.trans, Eq, MeasureTheory.lintegral, MeasureTheory.Measure.restrict, HDiv.hDiv, rfl, Eq.symm, MeasureTheory.setLIntegral_const, MeasureTheory.setLIntegral_mono', Eq.mpr, id, congrArg, MeasureTheory.lintegral_const_mul', ENNReal.coe_ne_top, MeasurableSpace, Eq.rec, Eq.refl, funext, ENNReal.ofReal, carlesonOperatorReal_mul and 34 more
153 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
lemma rcarleson_exceptional_set_estimate_specific TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 50 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, Measurable, LE.le, Norm.norm, Set, MeasurableSet, Set.Icc, HMul.hMul, Nat, Real.pi, Membership.mem, ENNReal, carlesonOperatorReal, K, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume and 5 more
Proof uses: Set.Ioo, HSub.hSub, Set.indicator, Eq, rfl, Eq.mpr, id, congrArg, norm_indicator_eq_indicator_norm, ite, Classical.decPred, Set.indicator.eq_1, dite, congr, ite_eq_left, mul_one, Not, ite_eq_right, of_eq_true, Eq.trans and 62 more
149 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
lemma rcarleson_exceptional_set_estimate_specific {δ C : ℝ≥0} (Cpos : 0 < C) {f : ℝ → ℂ}
  (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ C) {E : Set ℝ} (measurableSetE : MeasurableSet E)
  (E_subset : E ⊆ Set.Icc 0 (2 * π)) (hE : ∀ x ∈ E, δ ≤ T f x) :
    δ * volume E ≤ C * C10_0_1 4 2 * ENNReal.ofReal (2 * π + 2) ^ (2 : ℝ)⁻¹ * volume E ^ (2 : ℝ)⁻¹ := by
…
```
Mentions: E, T, C10_0_1
lemma rcarleson_exceptional_set_estimate_specific' TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 71 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, Measurable, LE.le, Norm.norm, Set, MeasurableSet, Set.Icc, HMul.hMul, Nat, Real.pi, Membership.mem, ENNReal, carlesonOperatorReal, K, HPow.hPow, MeasureTheory.Measure and 5 more
Proof uses: dite, Eq, Eq.mpr, id, congrArg, of_eq_true, Eq.trans, True, congrFun', ENNReal.zero_rpow_of_pos, MulZeroClass.mul_zero, Not, Ne, ne_of_gt, ENNReal.rpow_pos_of_nonneg, lt_of_le_of_ne', inferInstance, PartialOrder, Zero, zero_le and 28 more
146 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
```
lemma rcarleson_exceptional_set_estimate_specific' {δ C : ℝ≥0} (Cpos : 0 < C) {f : ℝ → ℂ}
  (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ C) {E : Set ℝ} (measurableSetE : MeasurableSet E)
  (E_subset : E ⊆ Set.Icc 0 (2 * π)) (hE : ∀ x ∈ E, δ ≤ T f x) :
    δ * volume E ^ (2 : ℝ)⁻¹ ≤ C * C10_0_1 4 2 * ENNReal.ofReal (2 * π + 2) ^ (2 : ℝ)⁻¹ := by
…
```
Mentions: E, T, C10_0_1
lemma rcarleson_exceptional_set_estimate_specific'' TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 90 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, Measurable, LE.le, Norm.norm, Set, MeasurableSet, Set.Icc, HMul.hMul, Nat, Real.pi, Membership.mem, ENNReal, carlesonOperatorReal, K, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume and 5 more
Proof uses: Eq.mpr, id, Eq, congrArg, propext, ENNReal.le_div_iff_mul_le, Or.inr, Ne, C10_0_1_pos, one_lt_two, ne_of_gt, lt_of_le_of_ne', mul_nonneg, le_of_lt, ENNReal.rpow_pos_of_nonneg, Iff.mpr, ENNReal.coe_pos, Mathlib.Meta.Positivity.pos_of_isNat, inferInstance, PartialOrder and 33 more
147 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
lemma rcarleson_exceptional_set_estimate_specific'' {δ C : ℝ≥0} (Cpos : 0 < C) {f : ℝ → ℂ}
  (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ C) {E : Set ℝ} (measurableSetE : MeasurableSet E)
  (E_subset : E ⊆ Set.Icc 0 (2 * π)) (hE : ∀ x ∈ E, δ ≤ T f x) :
    volume E ≤ C ^ (2 : ℝ) * C10_0_1 4 2 ^ (2 : ℝ) * ENNReal.ofReal (2 * π + 2) / δ ^ (2 : ℝ) := by
…
```
Mentions: E, T, C10_0_1
def C_distribution_carlesonOperatorReal_le' The constant used in C_distribution_carlesonOperatorReal_le' line 105 source
Source documentation
The constant used in C_distribution_carlesonOperatorReal_le'.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal
Proof uses: HPow.hPow, Real, HDiv.hDiv, HMul.hMul, C10_0_1, OfNat.ofNat, Nat, Real.toNNReal, HAdd.hAdd, Real.pi, Inv.inv
20 instances, coercions and auxiliary constants left out.
```
def C_distribution_carlesonOperatorReal_le' (δ ε : ℝ≥0) : ℝ≥0 :=
…
```
lemma C_distribution_carlesonOperatorReal_le'_pos TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 109 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, C_distribution_carlesonOperatorReal_le'
Proof uses: id, C10_0_1, Nat, C10_0_1_pos, one_lt_two, NNReal.rpow_pos, Inv.inv, Real, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.toNNReal, HAdd.hAdd, Real.pi, div_pos, inferInstance, PartialOrder, mul_pos, Iff.mpr, Real.toNNReal_pos and 8 more
75 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma C_distribution_carlesonOperatorReal_le'_pos {δ ε : ℝ≥0} (δpos : 0 < δ) (εpos : 0 < ε) :
    0 < C_distribution_carlesonOperatorReal_le' δ ε := by
…
```
Mentions: C_distribution_carlesonOperatorReal_le'
theorem C_distribution_carlesonOperatorReal_le'_property TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 115 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Eq, ENNReal, HDiv.hDiv, HMul.hMul, HPow.hPow, Real, C_distribution_carlesonOperatorReal_le', Nat, C10_0_1, ENNReal.ofReal, HAdd.hAdd, Real.pi
Proof uses: id, Eq.mpr, Real.toNNReal, Inv.inv, congrArg, ENNReal.coe_rpow_of_nonneg, of_eq_true, LE.le, Eq.trans, True, ENNReal.rpow_inv_rpow, Not, False, not_false_eq_true, ENNReal.coe_div, C10_0_1_pos, one_lt_two, ne_of_gt, div_pos, mul_pos and 41 more
144 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem C_distribution_carlesonOperatorReal_le'_property {δ ε : ℝ≥0} (δpos : 0 < δ) :
    ↑ε = ↑(C_distribution_carlesonOperatorReal_le' δ ε) ^ (2 : ℝ)
          * ↑(C10_0_1 4 2) ^ (2 : ℝ) * ENNReal.ofReal (2 * π + 2) / ↑δ ^ (2 : ℝ) := by
…
```
Mentions: C_distribution_carlesonOperatorReal_le', C10_0_1
lemma distribution_carlesonOperatorReal_le' TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 129 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, Measurable, LE.le, Norm.norm, C_distribution_carlesonOperatorReal_le', ENNReal, MeasureTheory.distribution, carlesonOperatorReal, K, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Ioc, HMul.hMul, Nat, Real.pi
Proof uses: Eq.mpr, MeasureTheory.Measure, Set, MeasureTheory.superlevelSet, id, Eq, congrArg, MeasureTheory.distribution_eq_measure_superlevelSet, Inter.inter, MeasureTheory.Measure.restrict_apply', measurableSet_Ioc, HDiv.hDiv, HPow.hPow, C10_0_1, ENNReal.ofReal, HAdd.hAdd, eq_of_heq, LE, HEq, Eq.refl and 26 more
117 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
lemma distribution_carlesonOperatorReal_le' {δ ε : ℝ≥0} (δpos : 0 < δ) (εpos : 0 < ε) {g : ℝ → ℂ}
  (hmg : Measurable g) (hg : ∀ x, ‖g x‖ ≤ C_distribution_carlesonOperatorReal_le' δ ε) :
    distribution (T g) δ (volume.restrict (Set.Ioc 0 (2 * π))) ≤ ε := by
…
```
Mentions: g, C_distribution_carlesonOperatorReal_le', T
def C_control_approximation_effect' The constant used in C_control_approximation_effect' line 151 source
Source documentation
The constant used in C_control_approximation_effect'.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal
Proof uses: Min.min, HDiv.hDiv, Real.toNNReal, HMul.hMul, Real, OfNat.ofNat, Real.pi, C_distribution_carlesonOperatorReal_le'
17 instances, coercions and auxiliary constants left out.
```
def C_control_approximation_effect' (δ ε : ℝ≥0) : ℝ≥0 :=
…
```
lemma C_control_approximation_effect'_pos TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 156 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, C_control_approximation_effect'
Proof uses: lt_min, HDiv.hDiv, Real.toNNReal, HMul.hMul, Real, Real.pi, C_distribution_carlesonOperatorReal_le', div_pos, Iff.mpr, Real.toNNReal_pos, mul_pos, inferInstance, PartialOrder, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, Nat and 3 more
54 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'_pos {δ ε : ℝ≥0} (δpos : 0 < δ) (εpos : 0 < ε) :
    0 < C_control_approximation_effect' δ ε :=
…
```
Mentions: C_control_approximation_effect'
lemma C_control_approximation_effect'_property TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 162 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LE.le, ENNReal, HMul.hMul, C_control_approximation_effect', ENNReal.ofReal, Real, OfNat.ofNat, Nat, Real.pi
Proof uses: id, Eq.mpr, Min.min, HDiv.hDiv, Real.toNNReal, C_distribution_carlesonOperatorReal_le', Eq, congrArg, ENNReal.coe_min, min_mul, ENNReal.coe_div, ne_of_gt, Iff.mpr, LT.lt, Real.toNNReal_pos, mul_pos, inferInstance, PartialOrder, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat and 16 more
74 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
lemma C_control_approximation_effect'_property {δ ε : ℝ≥0} :
    C_control_approximation_effect' δ ε * ENNReal.ofReal (2 * π) ≤ δ := by
…
```
Mentions: C_control_approximation_effect'
lemma C_control_approximation_effect'_le TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx line 170 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LE.le, C_control_approximation_effect', C_distribution_carlesonOperatorReal_le', Real.toNNReal, HDiv.hDiv, Real, HMul.hMul, OfNat.ofNat, Nat, Real.pi
Proof uses: min_le_right
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma C_control_approximation_effect'_le {δ ε : ℝ≥0} :
  C_control_approximation_effect' δ ε
    ≤ C_distribution_carlesonOperatorReal_le' (2 * π * (↑δ / 2) / 2).toNNReal (ε / 2) := by
…
```
Mentions: C_control_approximation_effect', C_distribution_carlesonOperatorReal_le'
lemma control_approximation_effect' TODO: can be removed once two_sided_metric_carleson_hasStrongType no longer depends on sorryAx This is Lemma 11.6.4 (partial Fourier sums of small) in the blueprint line 177 source
Source comment
TODO: can be removed once `two_sided_metric_carleson_hasStrongType` no longer depends on `sorryAx` This is Lemma 11.6.4 (partial Fourier sums of small) in the blueprint.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, LT.lt, OfNat.ofNat, Real, Complex, Measurable, Function.Periodic, HMul.hMul, Nat, Real.pi, LE.le, Norm.norm, C_control_approximation_effect', ENNReal, MeasureTheory.distribution, iSup, ENorm.enorm, partialFourierSum, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume and 1 more
Proof uses: Trans.trans, Eq, HAdd.hAdd, HDiv.hDiv, carlesonOperatorReal, K, Function.comp, RingHom, starRingEnd, ENNReal.ofReal, MeasureTheory.eLpNorm, operatorBound, rfl, MeasureTheory.distribution_mono_left, Eq.mpr, Filter.Eventually, MeasureTheory.ae, MeasureTheory.Measure, Membership.mem, Set and 100 more
231 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) (εpos : 0 < ε)
  {g : ℝ → ℂ} (g_measurable : Measurable g)
  (g_periodic : g.Periodic (2 * π))
  (g_bound : ∀ x, ‖g x‖ ≤ C_control_approximation_effect' δ ε) :
    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'