MainTheorem.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

3 declarations, 0 contain sorry, 0 declared as axioms

Theorem 10.0.1

Imports (2)

Imported by 3 files

def C10_0_1 The constant used in two_sided_metric_carleson line 15 source
Source documentation
The constant used in two_sided_metric_carleson. Has value 2 ^ (474 * a ^ 3) / (q - 1) ^ 6 in the blueprint.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, NNReal
Proof uses: HMul.hMul, HPow.hPow, C_K, Real, OfNat.ofNat, C1_0_2
11 instances, coercions and auxiliary constants left out.
```
def C10_0_1 (a : ℕ) (q : ℝ≥0) : ℝ≥0 := C_K a ^ 2 * C1_0_2 a q
```
Mentions: C_K, C1_0_2
lemma C10_0_1_pos line 17 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, NNReal, LT.lt, OfNat.ofNat, C10_0_1
Proof uses: mul_pos, HPow.hPow, C_K, Real, C1_0_2, pow_two_pos_of_ne_zero, id, Ne, Eq.mpr, Not, Eq, False, congrArg, eq_false, LT.lt.ne', C_K_pos, of_eq_true, not_false_eq_true, C1_0_2_pos
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem two_sided_metric_carleson Theorem 10.0.1 line 30 source
Source comment
Theorem 10.0.1

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, MetricSpace, MeasureTheory.DoublingMeasure, NNReal, defaultA, Set, Complex, IsTwoSidedKernel, CompatibleFunctions, Real, IsCancellative, defaultτ, LE.le, OfNat.ofNat, Membership.mem, Set.Ioc, NNReal.HolderConjugate, MeasurableSet, GT.gt, MeasureTheory.HasBoundedStrongType and 15 more
Proof uses: Neg.neg, LT.lt, Real.rpow_pos_of_pos, two_pos, IsOneSidedKernel, HSMul.hSMul, isOneSidedKernel_const_smul, id, abs, Eq.mpr, Eq, congrArg, neg_mul, Real.abs_rpow_of_nonneg, LT.lt.le, abs_two, Real.rpow_le_one_of_one_le_of_nonpos, one_le_two, of_eq_true, Eq.trans and 140 more
260 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
theorem two_sided_metric_carleson (ha : 4 ≤ a) (hq : q ∈ Ioc 1 2) (hqq' : q.HolderConjugate q')
    (hF : MeasurableSet F) (hG : MeasurableSet G)
    (hT : ∀ r > 0, HasBoundedStrongType (czOperator K r) 2 2 volume volume (C_Ts a))
    {f : X → ℂ} (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ F.indicator 1 x) :
    ∫⁻ x in G, carlesonOperator K f x ≤
    C10_0_1 a q * (volume G) ^ (q' : ℝ)⁻¹ * (volume F) ^ (q : ℝ)⁻¹ := by
…
```
Mentions: czOperator, K, C_Ts, carlesonOperator, C10_0_1