125.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 125

Reference: erdosproblems.com/125

There are four possibilities for the density of $A+B$:

$A+B$ has zero upper and lower density (and hence also zero density).
$A+B$ has zero lower density, but positive upper density (and hence no density).
$A+B$ has positive upper and lower density that are equal (and hence positive density).
$A+B$ has positive upper and lower density that are unequal (and hence no density).

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Erdos125.erdos_125 sorry line 66 source
depends on sorry what this means
Statement uses: Iff, False, Set.HasPosDensity, Nat, HAdd.hAdd, Set, Set.ofPred, LE.le, Finset, List.toFinset, Nat.digits, OfNat.ofNat, Insert.insert, Singleton.singleton, 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 Erdos125.erdos_125.variants.positive_lower_density sorry line 78 source
depends on sorry what this means
Statement uses: Iff, False, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, HAdd.hAdd, Set, Set.ofPred, LE.le, Finset, List.toFinset, Nat.digits, Insert.insert, Singleton.singleton, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos125.erdos_125.variants.positive_upper_density Literature question: Does $A + B$ have positive upper density? sorry line 88 source
Source documentation
Literature question: Does $A + B$ have positive upper density?

depends on sorry what this means
Statement uses: Iff, True, LT.lt, Real, OfNat.ofNat, Set.upperDensity, Nat, HAdd.hAdd, Set, Set.ofPred, LE.le, Finset, List.toFinset, Nat.digits, Insert.insert, Singleton.singleton, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_125.variants.positive_upper_density :
    answer(sorry) ↔ 0 < (A + B).upperDensity := by
…
```
theorem Erdos125.erdos_125.variants.zero_density Case 1: Does $A + B$ have zero upper and lower density? sorry line 97 source
Source documentation
Case 1: Does $A + B$ have zero upper and lower density?

depends on sorry what this means
Statement uses: Iff, True, And, Eq, Real, Set.upperDensity, Nat, HAdd.hAdd, Set, Set.ofPred, LE.le, Finset, List.toFinset, Nat.digits, OfNat.ofNat, Insert.insert, Singleton.singleton, Set.univ, Set.lowerDensity
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_125.variants.zero_density :
    answer(sorry) ↔ (A + B).upperDensity = 0 ∧ (A + B).lowerDensity = 0 := by
…
```
theorem Erdos125.erdos_125.variants.zero_lower_positive_upper_density Case 2: Does $A + B$ have zero lower density, but positive upper density? sorry line 106 source
Source documentation
Case 2: Does $A + B$ have zero lower density, but positive upper density?

depends on sorry what this means
Statement uses: Iff, True, And, Eq, Real, Set.lowerDensity, Nat, HAdd.hAdd, Set, Set.ofPred, LE.le, Finset, List.toFinset, Nat.digits, OfNat.ofNat, Insert.insert, Singleton.singleton, Set.univ, LT.lt, Set.upperDensity
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_125.variants.zero_lower_positive_upper_density :
    answer(sorry) ↔ (A + B).lowerDensity = 0 ∧ 0 < (A + B).upperDensity := by
…
```
theorem Erdos125.erdos_125.variants.positive_unequal_density sorry line 118 source
depends on sorry what this means
Statement uses: Iff, False, And, LT.lt, Real, OfNat.ofNat, Set.lowerDensity, Nat, HAdd.hAdd, Set, Set.ofPred, LE.le, Finset, List.toFinset, Nat.digits, Insert.insert, Singleton.singleton, Set.univ, Set.upperDensity
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

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