340.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

17 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 340

Reference: erdosproblems.com/340

Imports (2)

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Erdos340.greedySidon_go_singleton_two line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, And, GE.ge, OfNat.ofNat, Not, Membership.mem, Finset, Singleton.singleton, IsSidon, Union.union, Finset.greedySidon.go
Proof uses: Eq.mpr, LE.le, HAdd.hAdd, id, Eq.trans, LT.lt, Nat.find, DecidablePred, Exists, Eq.mpr_prop, Finset.Nonempty, True, eq_true, True.intro, funext, congr, congrArg, Set, Or, forall_congr and 31 more
87 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_go_pair_three line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, And, GE.ge, OfNat.ofNat, Not, Membership.mem, Finset, Insert.insert, Singleton.singleton, IsSidon, Union.union, Finset.greedySidon.go
Proof uses: Eq.mpr, LT.lt, LE.le, HAdd.hAdd, Or, id, Eq.trans, Nat.find, Eq.mpr_prop, Finset.Nonempty, True, eq_true, True.intro, congrFun', congrArg, Subtype, Subtype.mk, dite_congr, Exists, Eq.mpr_not and 33 more
78 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_zero line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: rfl
2 instances, coercions and auxiliary constants left out.
theorem Erdos340.greedySidon_one line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: Eq.mpr, Nat.find, And, GE.ge, HAdd.hAdd, Finset.max', Singleton.singleton, Finset, Iff.mp, Exists, Membership.mem, IsSidon, Prod.fst, Subtype, Finset.greedySidon.aux, Bool, Bool.true, Finset.Nonempty, Multiset, Finset.val and 57 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_two line 54 source
extra axioms: _private.0.Erdos340.greedySidon_two._native.decide.ax_1 what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_three line 58 source
extra axioms: _private.0.Erdos340.greedySidon_three._native.decide.ax_1 what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_four line 61 source
extra axioms: _private.0.Erdos340.greedySidon_four._native.decide.ax_1 what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_five line 65 source
extra axioms: _private.0.Erdos340.greedySidon_five._native.decide.ax_1 what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.greedySidon_ten line 69 source
extra axioms: _private.0.Erdos340.greedySidon_ten._native.decide.ax_1 what this means
Statement uses: Eq, Nat, Finset.greedySidon, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos340.erdos_340 Documented declaration: erdos_340 sorry line 80 source
Source documentation
Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the order of growth of $A$? Is it true that $|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$?

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, HDiv.hDiv, Real.sqrt, HPow.hPow, Set.ncard, Inter.inter, Set, Set.range, Finset.greedySidon, Set.Icc
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_340 (ε : ℝ) (hε : ε > 0) :
    (fun n : ℕ ↦ √n / n ^ ε) =O[atTop]
      fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ) := by
…
```
theorem Erdos340.erdos_340.variants.isTheta Documented declaration: isTheta sorry line 93 source
Source documentation
Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the order of growth of $A$? Is it true that $|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$?

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Asymptotics.IsTheta, Nat, Filter.atTop, Set.ncard, Inter.inter, Set, Set.range, Finset.greedySidon, Set.Icc, sorryAx, Bool.false
Proof uses: Lean.Name, 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_340.variants.isTheta (ε : ℝ) (hε : ε > 0) :
    (fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ)) =Θ[atTop]
      (answer(sorry) : ℕ → ℝ) := by
…
```
theorem Erdos340.erdos_340.variants.third It is trivial that this sequence grows at least like $\gg N^{1/3}$ sorry line 102 source
Source documentation
It is trivial that this sequence grows at least like $\gg N^{1/3}$.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, HPow.hPow, HDiv.hDiv, Set.ncard, Inter.inter, Set, Set.range, Finset.greedySidon, Set.Icc
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_340.variants.third (ε : ℝ) (hε : ε > 0) :
    (fun n : ℕ ↦ (n : ℝ) ^ ((1 : ℝ) / 3)) =O[atTop]
      fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ) := by
…
```
theorem Erdos340.erdos_340.variants.sub_hasPosDensity Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has positive density sorry line 115 source
Source documentation
Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has positive density.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

depends on sorry what this means
Statement uses: Set.HasPosDensity, Nat, HSub.hSub, Set, Set.range, Finset.greedySidon, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_340.variants.sub_hasPosDensity :
    Set.HasPosDensity (Set.range greedySidon - Set.range greedySidon) := by
…
```
Mentions: Set.HasPosDensity
theorem Erdos340.erdos_340.variants._22_mem_sub Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this contains $22$, which it does line 127 source
Source documentation
Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this contains $22$, which it does.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

extra axioms: _private.0.Erdos340.erdos_340.variants._22_mem_sub._native.decide.ax_1 what this means
Statement uses: Membership.mem, Nat, Set, HSub.hSub, Set.range, Finset.greedySidon, OfNat.ofNat
Proof uses: Eq, of_decide_eq_true, Eq.mpr, id, congrArg, Set.sub_mem_sub, Set.mem_range_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_340.variants._22_mem_sub :
    22 ∈ Set.range greedySidon - Set.range greedySidon := by
…
```
theorem Erdos340.erdos_340.variants._33_mem_sub The smallest integer which is unknown to be in $A - A$ is $33$ sorry line 137 source
Source documentation
The smallest integer which is unknown to be in $A - A$ is $33$.

depends on sorry what this means
Statement uses: Iff, True, Membership.mem, Nat, Set, HSub.hSub, Set.range, Finset.greedySidon, OfNat.ofNat
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 erdos_340.variants._33_mem_sub : answer(sorry) ↔
    33 ∈ Set.range greedySidon - Set.range greedySidon := by
…
```
theorem Erdos340.erdos_340.variants.cofinite_sub It may be true that all or almost all integers are in $A - A$ sorry line 149 source
Source documentation
It may be true that all or almost all integers are in $A - A$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Membership.mem, Set, HSub.hSub, Set.range, Finset.greedySidon, Filter.cofinite
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_340.variants.cofinite_sub : answer(sorry) ↔
    ∀ᶠ n in cofinite, n ∈ Set.range greedySidon - Set.range greedySidon := by
…
```
theorem Erdos340.erdos_340.variants.co_density_zero_sub It may be true that all or almost all integers are in $A - A$ sorry line 157 source
Source documentation
It may be true that all or almost all integers are in $A - A$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Set.HasDensity, OfNat.ofNat, Real, Set.univ, Membership.mem, Compl.compl, HSub.hSub, Set.range, Finset.greedySidon
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_340.variants.co_density_zero_sub : answer(sorry) ↔
    ∃ S : Set ℕ, S.HasDensity 0 ∧ ∀ n ∈ Sᶜ, n ∈ Set.range greedySidon - Set.range greedySidon := by
…
```