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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

8 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Carleson.ToMathlib.MeasureTheory.Function.LpSeminorm.Basic

Imported by 2 files

instance MeasureTheory.(instance at line 26) line 26 source
Source comment
TODO: move?
```
instance : ENormSMulClass ℝ≥0 ℝ≥0∞ where
…
```
theorem MeasureTheory.eLpNorm_le_eLpNorm_top_mul_eLpNorm' line 31 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENormedAddCommMonoid, MeasureTheory.Measure, ENNReal, MeasureTheory.AEStronglyMeasurable, NNReal, Filter.Eventually, LE.le, ENorm.enorm, HMul.hMul, MeasureTheory.ae, MeasureTheory.eLpNorm, Top.top
Proof uses: Trans.trans, HSMul.hSMul, MeasureTheory.eLpNorm_mono_enorm_ae, Eq.mpr, id, Eq, congrFun', congrArg, autoParam, funext, smul_mul_assoc, LE.le.trans, MeasureTheory.eLpNorm_const_smul_le'', enorm_NNReal, mul_assoc, mul_le_mul', le_refl, OfNat.ofNat, Or, LT.lt and 44 more
90 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
theorem MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_top'' line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENormedAddCommMonoid, MeasureTheory.Measure, ENNReal, MeasureTheory.AEStronglyMeasurable, NNReal, Filter.Eventually, LE.le, ENorm.enorm, HMul.hMul, MeasureTheory.ae, MeasureTheory.eLpNorm, Top.top
Proof uses: Trans.trans, Eq, MeasureTheory.eLpNorm_le_eLpNorm_top_mul_eLpNorm', flip, Eq.mpr, eq_of_heq, Filter, HEq, Eq.refl, HEq.refl, Eq.symm, funext, LE, of_eq_true, Eq.trans, True, congr, congrArg, mul_assoc, mul_comm and 2 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem MeasureTheory.eLpNorm'_le_eLpNorm'_mul_eLpNorm'' line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENormedAddCommMonoid, MeasureTheory.Measure, Real, MeasureTheory.AEStronglyMeasurable, NNReal, Filter.Eventually, LE.le, ENNReal, ENorm.enorm, HMul.hMul, MeasureTheory.ae, LT.lt, OfNat.ofNat, Eq, HDiv.hDiv, HAdd.hAdd, MeasureTheory.eLpNorm'
Proof uses: Trans.trans, HSMul.hSMul, id, HPow.hPow, MeasureTheory.lintegral, ENNReal.rpow_le_rpow, MeasureTheory.lintegral_mono_ae, Filter.Eventually.mono, Eq.mpr, congrArg, congrFun', Algebra.smul_mul_assoc, ENNReal.smul_def, smul_eq_mul, Eq.symm, mul_assoc, le_of_lt, div_pos, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat and 14 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm Hölder's inequality, as an inequality on the ℒp seminorm of an elementwise operation fun x => b (f x) (g x) line 97 source
Source documentation
Hölder's inequality, as an inequality on the ℒp seminorm of an elementwise operation fun x => b (f x) (g x).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENormedAddCommMonoid, MeasureTheory.Measure, ENNReal, MeasureTheory.AEStronglyMeasurable, NNReal, Filter.Eventually, LE.le, ENorm.enorm, HMul.hMul, MeasureTheory.ae, ENNReal.HolderTriple, MeasureTheory.eLpNorm
Proof uses: Eq, HDiv.hDiv, OfNat.ofNat, HAdd.hAdd, ENNReal.HolderTriple.one_div_eq, Or, Top.top, LT.lt, Real, ENNReal.toReal, ENNReal.trichotomy, of_eq_true, Eq.trans, True, congr, congrArg, congrFun', autoParam, id, Eq.mp and 41 more
113 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
```
theorem eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm {p q r : ℝ≥0∞}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (b : E → F → G) (c : ℝ≥0)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖ₑ ≤ c * ‖f x‖ₑ * ‖g x‖ₑ) [hpqr : HolderTriple p q r] :
    eLpNorm (fun x => b (f x) (g x)) r μ ≤ c * eLpNorm f p μ * eLpNorm g q μ := by
…
```
Mentions: g, E, c
theorem MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm' Hölder's inequality, as an inequality on the ℒp seminorm for functions to ℝ≥0∞ line 125 source
Source documentation
Hölder's inequality, as an inequality on the ℒp seminorm for functions to ℝ≥0∞.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, MeasureTheory.AEStronglyMeasurable, ENNReal.HolderTriple, LE.le, MeasureTheory.eLpNorm, HMul.hMul
Proof uses: LE.le.trans, OfNat.ofNat, NNReal, MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm, of_eq_true, Filter.Eventually, ENorm.enorm, MeasureTheory.ae, Eq.trans, True, congrFun', Filter, congrArg, funext, one_mul
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm' {p q r : ℝ≥0∞} {f g : α → ℝ≥0∞}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hpqr : HolderTriple p q r) :
    eLpNorm (fun x => f x * g x) r μ ≤ eLpNorm f p μ * eLpNorm g q μ := by
…
```
Mentions: g
theorem MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm'_of_enorm Hölder's inequality, as an inequality on the ℒp seminorm of an elementwise operation fun x => b (f x) (g x) line 133 source
Source documentation
Hölder's inequality, as an inequality on the ℒp seminorm of an elementwise operation fun x => b (f x) (g x).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENormedAddCommMonoid, MeasureTheory.Measure, ENNReal, MeasureTheory.AEStronglyMeasurable, NNReal, Filter.Eventually, LE.le, ENorm.enorm, HMul.hMul, MeasureTheory.ae, ENNReal.HolderTriple, MeasureTheory.eLpNorm
Proof uses: MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm
26 instances, coercions and auxiliary constants left out.
```
theorem eLpNorm_le_eLpNorm_mul_eLpNorm'_of_enorm {p q r : ℝ≥0∞} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (b : E → F → G) (c : ℝ≥0)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖ₑ ≤ c * ‖f x‖ₑ * ‖g x‖ₑ) [hpqr : HolderTriple p q r] :
    eLpNorm (fun x => b (f x) (g x)) r μ ≤ c * eLpNorm f p μ * eLpNorm g q μ :=
…
```
Mentions: g, E, c
theorem MeasureTheory.MemLp.of_bilin' line 140 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENormedAddCommMonoid, MeasureTheory.Measure, ENNReal, NNReal, MeasureTheory.MemLp, MeasureTheory.AEStronglyMeasurable, Filter.Eventually, LE.le, ENorm.enorm, HMul.hMul, MeasureTheory.ae, ENNReal.HolderTriple
Proof uses: And.intro, LT.lt, MeasureTheory.eLpNorm, Top.top, LE.le.trans_lt, MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_enorm, And.left, And.right, Ne.lt_top, ENNReal.mul_ne_top, ENNReal.coe_ne_top, MeasureTheory.MemLp.eLpNorm_ne_top
31 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.