CarlesonOperatorReal.lean`

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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

12 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Carleson.Classical.HilbertKernel

Imported by 2 files

def carlesonOperatorReal TODO: avoid this extra definition? line 15 source
Source comment
TODO: avoid this extra definition?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Complex, ENNReal
Proof uses: iSup, Int, LT.lt, OfNat.ofNat, ENorm.enorm, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.ofPred, Membership.mem, Set, Set.Ioo, Dist.dist, HMul.hMul, Complex.exp, Complex.I
40 instances, coercions and auxiliary constants left out.
```
def carlesonOperatorReal (K : ℝ → ℝ → ℂ) (f : ℝ → ℂ) (x : ℝ) : ENNReal :=
…
```
Mentions: K
lemma annulus_real_eq line 20 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, Eq, Set, Set.ofPred, Membership.mem, Set.Ioo, Dist.dist, Union.union, HSub.hSub, HAdd.hAdd
Proof uses: Set.ext, Eq.mpr, Iff, And, Or, LT.lt, id, congr, congrArg, congrFun', funext, abs, Eq.trans, Neg.neg, neg_sub, Iff.intro, Or.inl, And.intro, lt_of_not_ge, Mathlib.Tactic.Linarith.lt_irrefl and 63 more
115 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma annulus_real_volume line 32 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Membership.mem, Set, Set.Icc, OfNat.ofNat, Eq, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, Set.ofPred, Set.Ioo, Dist.dist, ENNReal.ofReal, HMul.hMul, Nat, HSub.hSub
Proof uses: Eq.mpr, Union.union, HAdd.hAdd, id, congrArg, annulus_real_eq, And.left, LE.le, MeasureTheory.measure_union, Disjoint, Inter.inter, EmptyCollection.emptyCollection, propext, Set.disjoint_iff, False.elim, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, LT.lt, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr and 74 more
122 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma annulus_measurableSet line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, MeasurableSet, Set.ofPred, Membership.mem, Set, Set.Ioo, Dist.dist
Proof uses: measurableSet_preimage, Measurable.dist, measurable_const, measurable_id, measurableSet_Ioo
20 instances, coercions and auxiliary constants left out.
lemma sup_eq_sup_dense_of_continuous line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, ENNReal, Set, IsOpen, Dense, ContinuousOn, Eq, iSup, Membership.mem, Inter.inter
Proof uses: le_antisymm, le_of_forall_lt_imp_le_of_dense, LT.lt, Exists, LE.le, Eq.mp, congrArg, propext, lt_iSup_iff, Set.preimage, Set.Ioi, ContinuousOn.isOpen_inter_preimage, isOpen_Ioi, Set.Nonempty, Dense.inter_open_nonempty, Exists.intro, And.intro, Eq.mpr, id, Eq.trans and 11 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma measurable_mul_kernel line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Real, Complex, Measurable, Prod, Function.uncurry, HMul.hMul, K, Complex.exp, Complex.I
Proof uses: Measurable.mul, Function.comp, Prod.snd, Measurable.comp, measurable_snd, Hilbert_kernel_measurable, Measurable.cexp, measurable_const, Complex.measurable_ofReal
39 instances, coercions and auxiliary constants left out.
def Qᵣ Rationals as a set of real numbers line 63 source
Source documentation
Rationals as a set of real numbers.
```
private def Qᵣ : Set ℝ := Rat.cast '' Set.univ
```
lemma Qᵣ_dense Rationals are dense in reals line 66 source
Source documentation
Rationals are dense in reals.
```
private lemma Qᵣ_dense : Dense Qᵣ :=
…
```
Mentions: Qᵣ
lemma Qᵣ_countable Rationals are countable after conversion to reals, too line 70 source
Source documentation
Rationals are countable after conversion to reals, too.
```
private lemma Qᵣ_countable : Countable Qᵣ :=
…
```
Mentions: Qᵣ
lemma carlesonOperatorReal_measurable line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Complex, MeasureTheory.AEStronglyMeasurable, MeasureTheory.MeasureSpace.volume, MeasureTheory.IntegrableOn, Set.Ioo, HAdd.hAdd, OfNat.ofNat, Nat, Measurable, ENNReal, carlesonOperatorReal, K
Proof uses: Not, Or, Classical.em, Measurable.iSup, Int, iSup, LT.lt, ENorm.enorm, MeasureTheory.integral, MeasureTheory.Measure.restrict, Set.ofPred, Membership.mem, Set, Dist.dist, HMul.hMul, Complex.exp, Complex.I, Set.indicator, Eq, rfl and 279 more
329 instances, coercions and auxiliary constants left out.

Tactic trace: 185 steps
Open to load the checked before-and-after goal states.
lemma carlesonOperatorReal_mul line 262 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Complex, LT.lt, OfNat.ofNat, Eq, ENNReal, carlesonOperatorReal, K, HMul.hMul, ENNReal.ofReal, HDiv.hDiv
Proof uses: Eq.mpr, iSup, Int, ENorm.enorm, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.ofPred, Membership.mem, Set, Set.Ioo, Dist.dist, Complex.exp, Complex.I, id, congrArg, carlesonOperatorReal.eq_1, ENNReal.mul_iSup, SupSet, Eq.rec and 24 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma carlesonOperatorReal_eq_of_restrict_interval line 283 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Complex, Membership.mem, Set, Set.Icc, Eq, ENNReal, carlesonOperatorReal, K, Set.indicator, Set.Ioo, HSub.hSub, OfNat.ofNat, HAdd.hAdd
Proof uses: id, SupSet, Eq.rec, iSup, Eq.refl, Int, LT.lt, ENorm.enorm, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.ofPred, Dist.dist, HMul.hMul, Complex.exp, Complex.I, funext, ENorm, NormedAddCommGroup, NormedSpace and 73 more
118 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.