NNReal.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.Analysis.Normed.Ring.Basic

Imported by 2 files

lemma NNReal.div_self_eq_ite MR wonders: is this lemma actually used? line 12 source
Source comment
MR wonders: is this lemma actually used? could or should it be golfed away?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, Eq, HDiv.hDiv, ite, LT.lt, OfNat.ofNat
Proof uses: dite, Eq.mpr, id, congrArg, ite_eq_left, div_self, LT.lt.ne', Not, ite_eq_right, of_eq_true, Eq.trans, True, congrFun', congr, Eq.mp, LE.le, div_zero, eq_self
36 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma div_self_eq_ite {x : ℝ≥0} : x / x = if 0 < x then 1 else 0 := by
…
```
lemma Real.nnnorm_le_nnnorm Transfer an inequality over ℝ to one of NNNorms over ℝ≥0 line 21 source
Source documentation
Transfer an inequality over ℝ to one of NNNorms over ℝ≥0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, NNReal, NNNorm.nnnorm
Proof uses: Eq.mpr, NNReal.mk, id, Eq, congrArg, Real.nnnorm_of_nonneg, LE.le.trans
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma Real.nnnorm_le_nnnorm {x y : ℝ} (hx : 0 ≤ x) (hy : x ≤ y) : ‖x‖₊ ≤ ‖y‖₊ := by
…
```