226.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 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, 1 contain sorry, 0 declared as axioms

Erdős Problem 226

References:

erdosproblems.com/226
[BaSc70] Barth, K. F. and Schneider, W. J., Entire functions mapping countable dense subsets of the reals onto each other monotonically. J. London Math. Soc. (2) (1970), 620--626.
[BaSc71] Barth, K. F. and Schneider, W. J., Entire functions mapping arbitrary countable dense sets and their complements onto each other. J. London Math. Soc. (2) (1971/72), 482--488.
[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.

Imports (1)

FormalConjecturesUtil

def Erdos226.PreservesRationality A real function preserves rationality line 40 source
Source documentation
A real function preserves rationality.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Iff, Membership.mem, Set, Set.range, Rat
3 instances, coercions and auxiliary constants left out.
```
def PreservesRationality (f : ℝ → ℝ) : Prop :=
…
```
theorem Erdos226.erdos_226 sorry line 51 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Complex, And, Differentiable, Real, Eq, Complex.im, OfNat.ofNat, AffineMap, Ne, Complex.re, Erdos226.PreservesRationality
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
63 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.