12.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

10 declarations, 9 contain sorry, 0 declared as axioms

Erdős Problem 12

Reference: erdosproblems.com/12

Imports (1)

FormalConjecturesUtil

Imported by 2 files

abbrev Erdos12.IsGood A set A is "good" if it is infinite and there are no distinct a,b,c in A such that a ∣ (b+c) and b > a, c > a line 36 source
Source documentation
A set A is "good" if it is infinite and there are no distinct a,b,c in A such that a ∣ (b+c) and b > a, c > a.

no axioms what this means
Statement uses: Set, Nat
Proof uses: And, Set.Infinite, Membership.mem, Dvd.dvd, HAdd.hAdd, LT.lt, Eq
5 instances, coercions and auxiliary constants left out.
```
abbrev IsGood (A : Set ℕ) : Prop := A.Infinite ∧
…
```
theorem Erdos12.isGood_example sorry line 45 source
depends on sorry what this means
Statement uses: Erdos12.IsGood, Set.ofPred, Nat, Exists, Nat.ModEq, OfNat.ofNat, Nat.Prime, Eq, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos12.erdos_12.parts.i sorry line 61 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Erdos12.IsGood, LT.lt, Real, OfNat.ofNat, Filter.liminf, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos12.erdos_12.parts.ii sorry line 76 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Set, Nat, Erdos12.IsGood, Set.Infinite, Set.ofPred, LT.lt, Set.ncard, Inter.inter, Set.Icc, HPow.hPow, HSub.hSub
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 Erdos12.erdos_12.parts.iii Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ sorry line 85 source
Source documentation
Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \in A} \frac{1}{n} < \infty$?

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Erdos12.IsGood, Summable, Real, HDiv.hDiv, OfNat.ofNat, Membership.mem, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_12.parts.iii :
    answer(sorry) ↔ ∀ (A : Set ℕ), IsGood A → Summable (fun (n : A) ↦ (1 / n : ℝ)) := by
…
```
Mentions: Erdos12.IsGood
theorem Erdos12.erdos_12.variants.erdos_sarkozy_density_0 Erdős and Sárközy proved that such an $A$ must have density 0 sorry line 95 source
Source documentation
Erdős and Sárközy proved that such an $A$ must have density 0. [ErSa70] Erd\H os, P. and Sárk"ozi, A., On the divisibility properties of sequences of integers. Proc. London Math. Soc. (3) (1970), 97-101

depends on sorry what this means
Statement uses: Set, Nat, Erdos12.IsGood, Set.HasDensity, OfNat.ofNat, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_12.variants.erdos_sarkozy_density_0 (A : Set ℕ) (hA : IsGood A) : A.HasDensity 0 := by
…
```
Mentions: Erdos12.IsGood
theorem Erdos12.erdos_12.variants.erdos_sarkozy Given any function sorry line 104 source
Source documentation
Given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$, such that there are infinitely many $N$ such that $$\lvert A\cap{1,\ldots,N}\rvert > \frac{N}{f(N)}.

depends on sorry what this means
Statement uses: Nat, Filter.Tendsto, Filter.atTop, Exists, Set, And, Erdos12.IsGood, Set.Infinite, Set.ofPred, LT.lt, Real, HDiv.hDiv, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_12.variants.erdos_sarkozy (f : ℕ → ℕ) (hf : atTop.Tendsto f atTop) :
    ∃ A, IsGood A ∧ {N : ℕ | (N : ℝ) / f N < (A ∩ Icc 1 N).ncard}.Infinite := by
…
```
Mentions: Erdos12.IsGood
theorem Erdos12.erdos_12.variants.example Documented declaration: example sorry line 115 source
Source documentation
An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ and such that $$\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N > 0$$ is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.

depends on sorry what this means
Statement uses: Set, Nat, Eq, Set.ofPred, Exists, Nat.Prime, Nat.ModEq, OfNat.ofNat, HPow.hPow, And, Erdos12.IsGood, LT.lt, Real, Filter.liminf, HDiv.hDiv, HMul.hMul, Set.ncard, Inter.inter, Set.Icc, Real.log and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_12.variants.example (A : Set ℕ)
    (hA : A = {p ^ 2 | (p : ℕ) (_ : p.Prime) (_ : p ≡ 3 [MOD 4])}) :
    IsGood A ∧ 0 < atTop.liminf (fun (N : ℕ) ↦ (A ∩ Icc 1 N).ncard * (N : ℝ).log / √N) := by
…
```
Mentions: Erdos12.IsGood
theorem Erdos12.erdos_12.variants.schoen Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ sorry line 128 source
Source documentation
Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then $$\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}$$ for infinitely many $N$.

depends on sorry what this means
Statement uses: Set, Nat, Erdos12.IsGood, Set.Pairwise, Nat.Coprime, Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Set.ofPred, LE.le, Set.ncard, Inter.inter, Set.Icc, HMul.hMul, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_12.variants.schoen (A : Set ℕ) (hA : IsGood A) (hA' : A.Pairwise Nat.Coprime) :
    ∃ C > (0 : ℝ), {N : ℕ | (A ∩ Icc 1 N).ncard ≤ C * (N : ℝ) ^ (2 / 3 : ℝ)}.Infinite := by
…
```
Mentions: Erdos12.IsGood
theorem Erdos12.erdos_12.variants.baier Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ sorry line 139 source
Source documentation
Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then $$\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}/\log N$$ for infinitely many $N$.

depends on sorry what this means
Statement uses: Set, Nat, Erdos12.IsGood, Set.Pairwise, Nat.Coprime, Exists, Real, And, GT.gt, OfNat.ofNat, Set.Infinite, Set.ofPred, LE.le, Set.ncard, Inter.inter, Set.Icc, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_12.variants.baier (A : Set ℕ) (hA : IsGood A) (hA' : A.Pairwise Nat.Coprime) :
    ∃ C > (0 : ℝ),
      {N : ℕ | (A ∩ Icc 1 N).ncard ≤ C * (N : ℝ) ^ (2 / 3 : ℝ) / (N : ℝ).log}.Infinite := by
…
```
Mentions: Erdos12.IsGood