NearestInt.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 7190 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, Formal Conjectures paper, arXiv v1

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def distToNearestInt The distance from a real number to the nearest integer line 25 source
Source documentation
The distance from a real number to the nearest integer.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: abs, HSub.hSub, round
9 instances, coercions and auxiliary constants left out.
```
noncomputable def distToNearestInt (x : ℝ) : ℝ := |x - round x|
…
```
theorem distToNearestInt_nonneg The distance to the nearest integer is nonnegative line 29 source
Source documentation
The distance to the nearest integer is nonnegative.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, distToNearestInt
Proof uses: abs_nonneg, HSub.hSub, round
22 instances, coercions and auxiliary constants left out.
```
theorem distToNearestInt_nonneg (x : ℝ) : 0 ≤ distToNearestInt x := abs_nonneg _
```
Mentions: distToNearestInt