CarlesonOnTheRealLineBasic.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

21 declarations, 0 contain sorry, 0 declared as axioms

Imports (4)

Imported by 4 files

lemma localOscillation_on_emptyset line 20 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, ContinuousMap, Real, Eq, ENNReal, localOscillation, EmptyCollection.emptyCollection, Set, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, iSup, Prod, Membership.mem, SProd.sprod, ENNReal.ofReal, Norm.norm, HAdd.hAdd, HSub.hSub, Prod.fst, Prod.snd, funext, Bot.bot, False, abs, iSup_congr_Prop and 9 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma localOscillation_on_empty_ball line 23 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, ContinuousMap, Real, LE.le, OfNat.ofNat, Eq, ENNReal, localOscillation, Metric.ball
Proof uses: Eq.mpr, EmptyCollection.emptyCollection, Set, id, congrArg, Iff.mpr, Metric.ball_eq_empty, localOscillation_on_emptyset, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def integer_linear line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, ContinuousMap, Real
Proof uses: ContinuousMap.mk, HMul.hMul
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
instance instFunctionDistancesReal line 131 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FunctionDistances, Real
Proof uses: FunctionDistances.mk, Int, integer_linear, Eq, ContinuousMap, PseudoMetricSpace.mk, Dist.mk, HMul.hMul, OfNat.ofNat, Max.max, abs, HSub.hSub, NNReal.mk, UniformSpace.ofDist, Bornology.ofDist
37 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma dist_integer_linear_eq line 149 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FunctionDistances.Θ, Real, Eq, Dist.dist, WithFunctionDistance, HMul.hMul, OfNat.ofNat, Nat, Max.max, abs, HSub.hSub
Proof uses: Eq.mpr, Int, id, Eq.trans, congrArg, congrFun', Eq.symm, Nat.cast_zero, Eq.refl
65 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma coeΘ_R line 153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FunctionDistances.Θ, Real, Eq, HMul.hMul
Proof uses: rfl
20 instances, coercions and auxiliary constants left out.
lemma coeΘ_R_C line 155 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FunctionDistances.Θ, Real, Eq, Complex, HMul.hMul
Proof uses: Eq.mpr, id, Eq.trans, congrArg, Eq.refl
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma oscillation_control line 157 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, FunctionDistances.Θ, LE.le, ENNReal, localOscillation, Metric.ball, FunctionDistances.coeΘ, ENNReal.ofReal, Dist.dist, WithFunctionDistance
Proof uses: dite, OfNat.ofNat, Eq.mpr, EmptyCollection.emptyCollection, Set, id, Eq, congrArg, Iff.mpr, Metric.ball_eq_empty, of_eq_true, Eq.trans, True, congrFun', iSup, Prod, Membership.mem, SProd.sprod, Norm.norm, HAdd.hAdd and 115 more
157 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
lemma frequency_monotone line 186 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, FunctionDistances.Θ, LE.le, Set, Metric.ball, Dist.dist, WithFunctionDistance
Proof uses: Eq.mpr, HMul.hMul, OfNat.ofNat, Nat, Max.max, abs, HSub.hSub, id, Eq, congrArg, dist_integer_linear_eq, dite, max_eq_right, mul_le_mul_of_nonneg_right, mul_le_mul_of_nonneg_left, le_max_right, le_of_lt, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 76 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma frequency_ball_doubling line 197 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, FunctionDistances.Θ, LE.le, Dist.dist, WithFunctionDistance, HMul.hMul, OfNat.ofNat, Nat
Proof uses: Eq.mpr, Max.max, abs, HSub.hSub, id, Eq, congrArg, dist_integer_linear_eq, dite, GE.ge, max_eq_left, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, HPow.hPow, Nat.rawCast and 66 more
98 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
theorem frequency_ball_growth line 208 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, FunctionDistances.Θ, LE.le, HMul.hMul, OfNat.ofNat, Nat, Dist.dist, WithFunctionDistance
Proof uses: Eq.mpr, Max.max, abs, HSub.hSub, id, Eq, congrArg, dist_integer_linear_eq, dite, GE.ge, max_eq_left, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, HPow.hPow, Nat.rawCast and 66 more
98 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma integer_ball_cover line 220 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, WithFunctionDistance, CoveredByBalls, Metric.ball, HMul.hMul, OfNat.ofNat, Nat
Proof uses: Eq.mpr, Exists, Finset, And, LE.le, Finset.card, Set, Set.iUnion, Membership.mem, id, Eq, congrArg, propext, coveredByBalls_iff, dite, GE.ge, Exists.intro, Singleton.singleton, FunctionDistances.Θ, And.intro and 215 more
239 instances, coercions and auxiliary constants left out.

Tactic trace: 147 steps
Open to load the checked before-and-after goal states.
instance compatibleFunctions_R line 363 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CompatibleFunctions, Real, HPow.hPow, Nat, OfNat.ofNat
Proof uses: CompatibleFunctions.mk, FunctionDistances.Θ, oscillation_control, frequency_monotone
22 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
instance real_van_der_Corput line 388 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsCancellative, Real, HPow.hPow, Nat, OfNat.ofNat, defaultτ
Proof uses: isCancellative_of_norm_integral_exp_le, Complex, LT.lt, Ne, ENNReal, iLipENorm, Top.top, LE.le, Set, Function.support, Metric.ball, FunctionDistances.Θ, Eq.mpr, Norm.norm, MeasureTheory.integral, NNReal, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, HMul.hMul, Complex.exp and 174 more
231 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
lemma Real.vol_real_eq remove? line 434 source
Source comment
remove?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Real.vol, HMul.hMul, OfNat.ofNat, Nat, abs, HSub.hSub
Proof uses: Eq.mpr, MeasureTheory.Measure.real, MeasureTheory.MeasureSpace.volume, Metric.ball, Dist.dist, id, congrArg, Real.vol.eq_1, ENNReal.toReal, MeasureTheory.Measure, Set, ENNReal, MeasureTheory.measureReal_def, Real.dist_eq, ENNReal.ofReal, Real.volume_ball, ENNReal.toReal_ofReal, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl and 56 more
91 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma Real.vol_real_eq {x y : ℝ} : Real.vol x y = 2 * |x - y| := by
…
```
Mentions: Real.vol
lemma Real.vol_eq proof can be simplified with enorm lemma line 439 source
Source comment
proof can be simplified with enorm lemma

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, ENNReal, vol, HMul.hMul, OfNat.ofNat, Nat, ENorm.enorm, HSub.hSub
Proof uses: Eq.mpr, MeasureTheory.Measure, Set, MeasureTheory.MeasureSpace.volume, Metric.ball, Dist.dist, id, congrArg, vol.eq_1, ENNReal.ofReal, Real.volume_ball, ENNReal.ofReal_mul, le_of_lt, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Nat.ble, EDist.edist and 4 more
82 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma Real.vol_eq {x y : ℝ} : vol x y = 2 * ‖x - y‖ₑ := by
…
```
Mentions: Real.vol
lemma Real.vol_real_symm remove? line 444 source
Source comment
remove?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Real.vol
Proof uses: Eq.mpr, HMul.hMul, OfNat.ofNat, Nat, abs, HSub.hSub, id, congrArg, Real.vol_real_eq, abs_sub_comm, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma Real.vol_real_symm {x y : ℝ} : Real.vol x y = Real.vol y x := by
…
```
Mentions: Real.vol
lemma Real.vol_symm proof can be simplified with enorm lemma line 448 source
Source comment
proof can be simplified with enorm lemma

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, ENNReal, vol
Proof uses: Eq.mpr, HMul.hMul, OfNat.ofNat, Nat, ENorm.enorm, HSub.hSub, id, congr, congrArg, Real.vol_eq, ENNReal.ofReal, Norm.norm, of_eq_true, Eq.trans, True, norm_sub_rev, eq_self
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma Real.vol_symm {x y : ℝ} : vol x y = vol y x := by
…
```
Mentions: Real.vol
instance isOneSidedKernelHilbert line 451 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsOneSidedKernel, Real, OfNat.ofNat, Nat, K
Proof uses: IsOneSidedKernel.mk, Hilbert_kernel_measurable, Eq.mpr, LE.le, Norm.norm, Complex, HDiv.hDiv, C_K, Real.vol, MeasureTheory.Measure.real, MeasureTheory.MeasureSpace.volume, Metric.ball, Dist.dist, id, Eq, congrArg, Real.vol.eq_1, ENNReal.toReal, MeasureTheory.Measure, Set and 174 more
142 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
instance isTwoSidedKernelHilbert line 485 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsTwoSidedKernel, Real, OfNat.ofNat, Nat, K
Proof uses: IsTwoSidedKernel.mk, LE.le, HMul.hMul, Dist.dist, Eq.mpr, ENNReal, ENorm.enorm, Complex, HSub.hSub, HPow.hPow, HDiv.hDiv, EDist.edist, Inv.inv, C_K, vol, RingHom, starRingEnd, id, Eq, congrArg and 15 more
96 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma carlesonOperatorReal_le_carlesonOperator TODO : change name to reflect that this only holds for a specific instance of CarlesonOperaterReal? line 496 source
Source comment
TODO : change name to reflect that this only holds for a specific instance of CarlesonOperaterReal?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, Real, Complex, ENNReal, carlesonOperatorReal, K, carlesonOperator, HPow.hPow, Nat, OfNat.ofNat
Proof uses: Eq.mpr, iSup, FunctionDistances.Θ, linearizedCarlesonOperator, id, Eq, congrArg, carlesonOperator.eq_1, Int, LT.lt, ENorm.enorm, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.ofPred, Membership.mem, Set, Set.Ioo, Dist.dist, HMul.hMul and 47 more
91 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
lemma carlesonOperatorReal_le_carlesonOperator : T ≤ carlesonOperator K := by
…
```
Mentions: T, carlesonOperator, K