Misc.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

1 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

theorem lintegral_antitone_mul_le This is Theorem 4.19 in https://doi.org/10.1007/978-3-319-30034-4 line 13 source
Source comment
This is Theorem 4.19 in https://doi.org/10.1007/978-3-319-30034-4

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, ENNReal, AEMeasurable, MeasureTheory.MeasureSpace.volume, LE.le, MeasureTheory.lintegral, MeasureTheory.Measure.restrict, Set.Iio, Antitone, HMul.hMul
Proof uses: MeasureTheory.Antitone.ennreal_induction', MeasureTheory.SimpleFunc.antitone_induction, MeasureTheory.SimpleFunc, Eq.mpr, MeasureTheory.SimpleFunc.restrict, MeasureTheory.SimpleFunc.const, Set.indicator, id, Eq, congr, congrArg, funext, congrFun', MeasurableSpace, Zero, LinearOrder, TopologicalSpace, OpensMeasurableSpace, ClosedIciTopology, MeasureTheory.SimpleFunc.restrict_apply and 72 more
165 instances, coercions and auxiliary constants left out.

Tactic trace: 112 steps
Open to load the checked before-and-after goal states.
```
theorem lintegral_antitone_mul_le {f g k : ℝ≥0 → ℝ≥0∞} (hf : AEMeasurable f) (hg : AEMeasurable g)
  (h : ∀ {t}, ∫⁻ s in Set.Iio t, f s ≤ ∫⁻ s in Set.Iio t, g s) (hk : Antitone k) :
    ∫⁻ s, k s * f s ≤ ∫⁻ s, k s * g s := by
…
```
Mentions: g, k, s