ENorm.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

17 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 4 files

lemma enorm_toReal_le line 16 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ENNReal, LE.le, ENorm.enorm, Real, ENNReal.toReal
Proof uses: of_eq_true, Eq.trans, ENNReal.ofReal, True, congrFun', congrArg, Norm.norm, ENNReal.abs_toReal
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma enorm_NNReal line 18 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, Eq, ENNReal, ENorm.enorm
Proof uses: Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
class ENormedAddCommSubMonoid An enormed monoid is an additive monoid endowed with a continuous enorm line 27 source
Source documentation
An enormed monoid is an additive monoid endowed with a continuous enorm. Note: not sure if this is the "right" class to add to Mathlib.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
class ENormedAddCommSubMonoid (E : Type*) [TopologicalSpace E] extends ENormedAddCommMonoid E, Sub E where
…
```
Mentions: E
instance (instance at line 49) line 49 source
instance (instance at line 52) line 52 source
instance (instance at line 57) line 57 source
Source comment
TODO: the generated instance name has suffix `_carleson`, not 100% sure why
```
instance : ContinuousENorm ℝ≥0 where
…
```
instance (instance at line 62) line 62 source
Source comment
TODO: the generated instance name has suffix `_carleson`, not 100% sure why
```
instance : ENormedAddCommMonoid ℝ≥0 where
…
```
instance (instance at line 72) line 72 source
instance (instance at line 75) line 75 source
lemma MeasureTheory.esub_zero unused line 82 source
Source comment
unused

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, ENormedAddCommSubMonoid, Eq, HSub.hSub, OfNat.ofNat
Proof uses: Eq.mpr, HAdd.hAdd, id, congrArg, Eq.symm, add_zero, ENormedAddCommSubMonoid.sub_add_cancel_of_enorm_le, LE.le, ENNReal, ENorm.enorm, congrFun', enorm_zero, of_eq_true
33 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma esub_zero [TopologicalSpace E] [ENormedAddCommSubMonoid E] {x : E} : x - 0 = x := by
…
```
Mentions: E, ENormedAddCommSubMonoid
theorem MeasureTheory._root_.ENNReal.toNNReal_smul TODO: find better place for this? line 90 source
Source comment
TODO: find better place for this?
```
theorem _root_.ENNReal.toNNReal_smul {α : Type*} {c : ℝ≥0∞} (hc : c ≠ ⊤) {f : α → ℝ≥0∞} :
    c.toNNReal • f = c • f := by
…
```
Mentions: c
theorem MeasureTheory.eLpNorm_const_nnreal_smul_le TODO: put next to MeasureTheory.eLpNorm_const_smul_le (which perhaps can stay) line 105 source
Source comment
TODO: put next to MeasureTheory.eLpNorm_const_smul_le (which perhaps can stay)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, ESeminormedAddMonoid, SMul, NNReal, ENormSMulClass, MeasurableSpace, ENNReal, MeasureTheory.Measure, LE.le, MeasureTheory.eLpNorm, HSMul.hSMul, HMul.hMul, ENorm.enorm
Proof uses: MeasureTheory.eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul', NNNorm.nnnorm, Filter.univ_mem', MeasureTheory.ae, Set.ofPred, id, Membership.mem, Set, le_of_eq, of_eq_true, Eq, Eq.trans, True, congrFun', congrArg, enorm_smul, enorm_NNReal, eq_self
31 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem eLpNorm_const_nnreal_smul_le
    {α : Type*} {m0 : MeasurableSpace α} {p : ℝ≥0∞}
    {μ : Measure α} {c : ℝ≥0} {f : α → ε} : eLpNorm (c • f) p μ ≤ ‖c‖ₑ * eLpNorm f p μ := by
…
```
Mentions: c
theorem MeasureTheory.eLpNorm_const_smul' TODO: put next to eLpNorm_const_smul line 112 source
Source comment
TODO: put next to eLpNorm_const_smul

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, ESeminormedAddCommMonoid, Module, NNReal, ENormSMulClass, MeasurableSpace, ENNReal, MeasureTheory.Measure, Eq, MeasureTheory.eLpNorm, HSMul.hSMul, HMul.hMul, ENorm.enorm
Proof uses: OfNat.ofNat, Ne, Or, eq_or_ne, of_eq_true, Eq.trans, True, congr, congrArg, congrFun', autoParam, zero_smul, MeasureTheory.eLpNorm_zero, enorm_zero, MulZeroClass.zero_mul, eq_self, Eq.symm, le_antisymm, MeasureTheory.eLpNorm_const_nnreal_smul_le, ENNReal.mul_le_of_le_div' and 14 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem eLpNorm_const_smul' {α : Type*} {m0 : MeasurableSpace α} {p : ℝ≥0∞}
    {μ : Measure α} {c : ℝ≥0} {f : α → ε'} :
    eLpNorm (c • f) p μ = ‖c‖ₑ * eLpNorm f p μ := by
…
```
Mentions: c
theorem MeasureTheory.eLpNorm_top_smul line 121 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, MeasureTheory.Measure, MeasureTheory.AEStronglyMeasurable, Eq, MeasureTheory.eLpNorm, HSMul.hSMul, Top.top, HMul.hMul
Proof uses: dite, OfNat.ofNat, of_eq_true, Eq.trans, True, congr, congrArg, congrFun', autoParam, MeasureTheory.eLpNorm_exponent_zero, MulZeroClass.mul_zero, eq_self, Not, Filter.EventuallyEq, MeasureTheory.ae, Classical.propDecidable, Eq.mpr, id, MeasureTheory.eLpNorm_eq_zero_of_ae_zero, Filter.mp_mem and 46 more
87 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
theorem MeasureTheory.eLpNorm_const_smul'' TODO: put next to eLpNorm_const_smul line 164 source
Source comment
TODO: put next to eLpNorm_const_smul

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, MeasureTheory.Measure, Ne, Top.top, Eq, MeasureTheory.eLpNorm, HSMul.hSMul, HMul.hMul
Proof uses: Eq.mpr, NNReal, ENNReal.toNNReal, id, congrArg, Eq.symm, ENNReal.toNNReal_smul, ENorm.enorm, MeasureTheory.eLpNorm_const_smul', HMul, Eq.rec, Eq.refl, of_eq_true, Eq.trans, True, congrFun', enorm_NNReal, ENNReal.coe_toNNReal, Not, False and 3 more
48 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem eLpNorm_const_smul'' {α : Type*} {m0 : MeasurableSpace α} {p : ℝ≥0∞}
  {μ : Measure α} {c : ℝ≥0∞} (hc : c ≠ ⊤) {f : α → ℝ≥0∞} :
    eLpNorm (c • f) p μ = c * eLpNorm f p μ := by
…
```
Mentions: c
theorem MeasureTheory.eLpNorm_const_smul''' TODO: put next to eLpNorm_const_smul line 172 source
Source comment
TODO: put next to eLpNorm_const_smul

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, MeasureTheory.Measure, MeasureTheory.AEStronglyMeasurable, Eq, MeasureTheory.eLpNorm, HSMul.hSMul, HMul.hMul
Proof uses: dite, Top.top, Eq.mpr, id, congr, congrArg, congrFun', autoParam, MeasureTheory.eLpNorm_top_smul, Not, MeasureTheory.eLpNorm_const_smul''
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem eLpNorm_const_smul''' {α : Type*} {m0 : MeasurableSpace α} {p : ℝ≥0∞}
  {μ : Measure α} {c : ℝ≥0∞} {f : α → ℝ≥0∞} (hf : AEStronglyMeasurable f μ) :
    eLpNorm (c • f) p μ = c * eLpNorm f p μ := by
…
```
Mentions: c
lemma MeasureTheory.eLpNormEssSup_const_nnreal_smul_le TODO: put next to the unprimed version; perhaps both should stay line 181 source
Source comment
TODO: put next to the unprimed version; perhaps both should stay

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, ESeminormedAddMonoid, SMul, NNReal, ENormSMulClass, MeasurableSpace, MeasureTheory.Measure, LE.le, ENNReal, MeasureTheory.eLpNormEssSup, HSMul.hSMul, HMul.hMul, ENorm.enorm
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, enorm_smul, enorm_NNReal, MeasureTheory.eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul', Filter.univ_mem', MeasureTheory.ae, Set.ofPred, id, Membership.mem, Set
32 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma eLpNormEssSup_const_nnreal_smul_le {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
    {c : ℝ≥0} {f : α → ε} : eLpNormEssSup (c • f) μ ≤ ‖c‖ₑ * eLpNormEssSup f μ := by
…
```
Mentions: c