9.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

6 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 9

Reference: erdosproblems.com/9

Imports (1)

FormalConjecturesUtil

def Erdos9.Erdos9A The set of odd numbers that cannot be expressed as a prime plus two powers of 2 line 33 source
Source documentation
The set of odd numbers that cannot be expressed as a prime plus two powers of 2.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, And, Odd, Not, Exists, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def Erdos9A : Set ℕ := { n | Odd n ∧ ¬ ∃ (p k l : ℕ), (Nat.Prime p) ∧ n = p + 2 ^ k + 2 ^ l }
…
```
theorem Erdos9.erdos9A_contains_one line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Membership.mem, Nat, Set, Erdos9.Erdos9A, OfNat.ofNat
Proof uses: And.intro, Odd, Not, Exists, And, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Ne, Eq.trans, binderNameHint, forall_congr and 76 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem Erdos9.erdos9A_contains_three line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Membership.mem, Nat, Set, Erdos9.Erdos9A, OfNat.ofNat
Proof uses: And.intro, Odd, Not, Exists, And, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Ne, Eq.trans, binderNameHint, forall_congr and 71 more
78 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem Erdos9.erdos9A_not_contains_five line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Membership.mem, Nat, Set, Erdos9.Erdos9A, OfNat.ofNat
Proof uses: id, Eq.mpr, Set.ofPred, And, Odd, Exists, Nat.Prime, Eq, HAdd.hAdd, HPow.hPow, Eq.trans, congrArg, congrFun', funext, forall_congr, implies_congr, Eq.refl, propext, exists_prop_congr, Iff.of_eq and 7 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Erdos9.erdos_9.variants.infinite The set is known to be infinite sorry line 68 source
Source documentation
The set is known to be infinite. In [Er77c] Erdős credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference.
[Er77c] Erdős, P., Problems and results on combinatorial number theory. III..

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Erdos9.Erdos9A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_9.variants.infinite : Erdos9A.Infinite := by
…
```
theorem Erdos9.erdos_9 Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive? sorry line 76 source
Source documentation
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?

depends on sorry what this means
Statement uses: Iff, True, LT.lt, Real, OfNat.ofNat, Set.upperDensity, Nat, Erdos9.Erdos9A, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_9 : answer(sorry) ↔ 0 < Erdos9A.upperDensity := by
…
```