Helper.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

8 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Imported by 3 files

lemma intervalIntegral.integral_conj' line 12 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasureTheory.Measure, Real, RCLike, Eq, intervalIntegral, RingHom, starRingEnd
Proof uses: Eq.mpr, HSMul.hSMul, ite, LE.le, OfNat.ofNat, Neg.neg, MeasureTheory.integral, MeasureTheory.Measure.restrict, Set.uIoc, id, congrArg, intervalIntegral.intervalIntegral_eq_integral_uIoc, integral_conj, HMul.hMul, RCLike.real_smul_eq_coe_mul, map_mul, RCLike.conj_ofReal, Eq.refl
79 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma intervalIntegrable_of_bdd line 17 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Complex, Measurable, LE.le, Norm.norm, IntervalIntegrable, MeasureTheory.MeasureSpace.volume
Proof uses: IntervalIntegrable.mono_fun', intervalIntegrable_const, Measurable.aestronglyMeasurable, MeasureTheory.Measure.restrict, Set.uIoc, Eq.mpr, Filter.EventuallyLE, MeasureTheory.ae, MeasureTheory.Measure, Filter.Eventually, id, Eq, congrArg, Filter.EventuallyLE.eq_1, Membership.mem, Set, Subtype, MeasureTheory.Measure.comap, propext, ae_restrict_iff_subtype and 4 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma IntervalIntegrable.bdd_mul line 27 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedDivisionRing, Real, MeasureTheory.Measure, IntervalIntegrable, MeasureTheory.AEStronglyMeasurable, Exists, LE.le, Norm.norm, HMul.hMul
Proof uses: Eq.mpr, MeasureTheory.IntegrableOn, Set.uIoc, id, Eq, congrArg, propext, intervalIntegrable_iff, MeasureTheory.Integrable, MeasureTheory.Measure.restrict, MeasureTheory.IntegrableOn.eq_1, MeasureTheory.Integrable.bdd_mul, Eq.symm, MeasureTheory.AEStronglyMeasurable.restrict, MeasureTheory.ae_of_all
51 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma IntervalIntegrable.mul_bdd line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedField, Real, MeasureTheory.Measure, IntervalIntegrable, MeasureTheory.AEStronglyMeasurable, Exists, LE.le, Norm.norm, HMul.hMul
Proof uses: Eq.mpr, id, Eq, congrFun', congrArg, funext, Eq.trans, mul_comm, Eq.refl, IntervalIntegrable.bdd_mul
49 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma IntegrableOn.sub line 41 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, NormedAddCommGroup, Set, MeasureTheory.IntegrableOn, HSub.hSub
Proof uses: MeasureTheory.Integrable.sub, MeasureTheory.Measure.restrict, Eq.mpr, MeasureTheory.Integrable, id, Eq, congrArg, Eq.symm, MeasureTheory.IntegrableOn.eq_1
18 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma ConditionallyCompleteLattice.le_biSup line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ConditionallyCompleteLinearOrder, Set, BddAbove, Set.image, Exists, And, Membership.mem, Eq, LE.le, iSup
Proof uses: le_csSup, Set.range, Eq.mpr, id, congrArg, propext, bddAbove_def, Eq.mp, Exists.intro, Max.max, SupSet.sSup, EmptyCollection.emptyCollection, dite, Classical.propDecidable, Singleton.singleton, Set.eq_singleton_iff_unique_mem, And.intro, Set.mem_range_self, Eq.symm, Eq.trans and 16 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.
theorem Function.Periodic.exists_mem_Ico₀' Adapted from mathlib Function.Periodic.exists_mem_Ico₀ line 82 source
Source comment
Adapted from mathlib Function.Periodic.exists_mem_Ico₀

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, LinearOrder, IsOrderedAddMonoid, Archimedean, Function.Periodic, LT.lt, OfNat.ofNat, Exists, Int, And, Membership.mem, Set, Set.Ico, HSub.hSub, HSMul.hSMul, Eq
Proof uses: ExistsUnique, LE.le, existsUnique_zsmul_near_of_pos', Exists.intro, And.intro, Eq.symm, Function.Periodic.sub_zsmul_eq
50 instances, coercions and auxiliary constants left out.
```
theorem Function.Periodic.exists_mem_Ico₀' {α β : Type*} {f : α → β} {c : α}
    [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] [Archimedean α]
    (h : Periodic f c) (hc : 0 < c) (x : α) :
    ∃ (n : ℤ), (x - n • c) ∈ Set.Ico 0 c ∧ f x = f (x - n • c) :=
…
```
Mentions: c
theorem Function.Periodic.exists_mem_Ico' Adapted from mathlib Function.Periodic.exists_mem_Ico₀ line 90 source
Source comment
Adapted from mathlib Function.Periodic.exists_mem_Ico₀

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, LinearOrder, IsOrderedAddMonoid, Archimedean, Function.Periodic, LT.lt, OfNat.ofNat, Exists, Int, And, Membership.mem, Set, Set.Ico, HAdd.hAdd, HSub.hSub, HSMul.hSMul, Eq
Proof uses: ExistsUnique, existsUnique_sub_zsmul_mem_Ico, Exists.intro, And.intro, Eq.symm, Function.Periodic.sub_zsmul_eq
55 instances, coercions and auxiliary constants left out.
```
theorem Function.Periodic.exists_mem_Ico' {α β : Type*} {f : α → β} {c : α}
    [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] [Archimedean α]
    (h : Periodic f c) (hc : 0 < c) (x a : α) :
    ∃ (n : ℤ), (x - n • c) ∈ Set.Ico a (a + c) ∧ f x = f (x - n • c) :=
…
```
Mentions: c