1136.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, 4 contain sorry, 0 declared as axioms

Erdős Problem 1136

References:

erdosproblems.com/1136
[Mu11] Müller, Helmut, Über ein additiv-zahlentheoretisches Problem von P. Erdős. Mitt. Math. Ges. Hamburg (2011), 75-78.

Imports (1)

FormalConjecturesUtil

def Erdos1136.AvoidsPowersOfTwo A set A of natural numbers has the property in the question if a + b ≠ 2 ^ k for all a, b ∈ A (not necessarily distinct) and all k ≥ 0 line 37 source
Source documentation
A set A of natural numbers has the property in the question if a + b ≠ 2 ^ k for all a, b ∈ A (not necessarily distinct) and all k ≥ 0.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Membership.mem, Ne, HAdd.hAdd, HPow.hPow, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def AvoidsPowersOfTwo (A : Set ℕ) : Prop :=
…
```
def Erdos1136.muellerSet The set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for some $i\geq 0$ line 41 source
Source documentation
The set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for some $i\geq 0$.

standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Nat.ModEq, HPow.hPow, OfNat.ofNat, HAdd.hAdd, HMul.hMul
9 instances, coercions and auxiliary constants left out.
```
def muellerSet : Set ℕ := {n | ∃ i : ℕ, n ≡ 3 * 2 ^ i [MOD 2 ^ (i + 2)]}
…
```
theorem Erdos1136.erdos_1136 Does there exist $A\subset \mathbb{N}$ with lower density $>1/3$ such that $a+b\neq 2^k$ for any $a,b\in A$ and $k\geq 0$? sorry line 52 source
Source documentation
Does there exist $A\subset \mathbb{N}$ with lower density $>1/3$ such that $a+b\neq 2^k$ for any $a,b\in A$ and $k\geq 0$?
Müller [Mu11] settled this question in the affirmative: in fact one can take $A$ to be the set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for any $i\geq 0$, which has density $1/2$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, LT.lt, Real, HDiv.hDiv, OfNat.ofNat, Set.lowerDensity, Set.univ, Erdos1136.AvoidsPowersOfTwo
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1136 : answer(True) ↔
    ∃ A : Set ℕ, (1 / 3 : ℝ) < A.lowerDensity ∧ AvoidsPowersOfTwo A := by
…
```
Mentions: Erdos1136.AvoidsPowersOfTwo
theorem Erdos1136.erdos_1136.variants.multiples_of_three Achieving density $1/3$ is trivial, taking $A$ to be all multiples of $3$ sorry line 60 source
Source documentation
Achieving density $1/3$ is trivial, taking $A$ to be all multiples of $3$.

depends on sorry what this means
Statement uses: And, Erdos1136.AvoidsPowersOfTwo, Set.ofPred, Nat, Dvd.dvd, OfNat.ofNat, Set.HasDensity, HDiv.hDiv, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1136.variants.multiples_of_three :
    AvoidsPowersOfTwo {n : ℕ | 3 ∣ n} ∧ Set.HasDensity {n : ℕ | 3 ∣ n} (1 / 3) := by
…
```
Mentions: Erdos1136.AvoidsPowersOfTwo, Set.HasDensity
theorem Erdos1136.erdos_1136.variants.mueller Müller [Mu11] settled this question in the affirmative: in fact one can take sorry line 70 source
Source documentation
Müller [Mu11] settled this question in the affirmative: in fact one can take $A$ to be the set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for any $i\geq 0$, which has density $1/2$.

depends on sorry what this means
Statement uses: And, Erdos1136.AvoidsPowersOfTwo, Erdos1136.muellerSet, Set.HasDensity, Nat, HDiv.hDiv, Real, OfNat.ofNat, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1136.variants.mueller :
    AvoidsPowersOfTwo muellerSet ∧ muellerSet.HasDensity (1 / 2) := by
…
```
Mentions: Erdos1136.AvoidsPowersOfTwo, Erdos1136.muellerSet
theorem Erdos1136.erdos_1136.variants.upper_bound Müller also proved this is best possible, in that $A$ with the property in the question has lower density at most $1/2$ sorry line 79 source
Source documentation
Müller also proved this is best possible, in that $A$ with the property in the question has lower density at most $1/2$.

depends on sorry what this means
Statement uses: Set, Nat, Erdos1136.AvoidsPowersOfTwo, LE.le, Real, Set.lowerDensity, Set.univ, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1136.variants.upper_bound (A : Set ℕ) (hA : AvoidsPowersOfTwo A) :
    A.lowerDensity ≤ 1 / 2 := by
…
```
Mentions: Erdos1136.AvoidsPowersOfTwo