39.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

12 declarations, 3 contain sorry, 0 declared as axioms

Green's Open Problem 39

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[BJR11] Bollobás, Béla, Svante Janson, and Oliver Riordan. "On covering by translates of a set." Random Structures & Algorithms 38.1‐2 (2011): 33-67.

Imports (2)

def Green39.proportionCoverable The proportion of subsets of $\mathbb{Z}/p\mathbb{Z}$ of size $k$ that can cover $\mathbb{Z}/p\mathbb{Z}$ using at most $c$ translates line 44 source
Source documentation
The proportion of subsets of $\mathbb{Z}/p\mathbb{Z}$ of size $k$ that can cover $\mathbb{Z}/p\mathbb{Z}$ using at most $c$ translates.
If p = 0 or k > p, return 0 by convention.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: dite, Eq, OfNat.ofNat, Not, ite, GT.gt, NeZero, Finset, ZMod, Finset.powersetCard, Finset.univ, Finset.filter, Exists, And, LE.le, Finset.card, HAdd.hAdd, HDiv.hDiv
27 instances, coercions and auxiliary constants left out.
```
def proportionCoverable (p k c : ℕ) : ℚ :=
…
```
Mentions: c
theorem Green39.proportionCoverable_p_p_1 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_t_0 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_2_1_2 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_3_1_2 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_a_gt_p line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_7_4_2 line 69 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat, HDiv.hDiv
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_11_3_4 line 74 source
extra axioms: _private.0.Green39.proportionCoverable_11_3_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat, HDiv.hDiv
Proof uses: of_decide_eq_true
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.proportionCoverable_11_4_3 line 79 source
extra axioms: _private.0.Green39.proportionCoverable_11_4_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Rat, Green39.proportionCoverable, OfNat.ofNat, Nat, HDiv.hDiv
Proof uses: of_decide_eq_true
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Green39.green_39 If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$? sorry line 88 source
Source documentation
If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$? [Gr24]

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Subtype, Nat, Nat.Prime, Real, Nat.sqrt, HMul.hMul, OfNat.ofNat, Green39.proportionCoverable, Filter.atTop, nhds
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 green_39 : answer(sorry) ↔
    Tendsto
      (fun p : {q : ℕ // q.Prime} ↦
        let k := Nat.sqrt p
…
```
theorem Green39.green_39.variant_101 "I do not know how to answer this even with 100 replaced by 1.01." [Gr24]" sorry line 101 source
Source documentation
"I do not know how to answer this even with 100 replaced by 1.01." [Gr24]"

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Subtype, Nat, Nat.Prime, Real, Nat.sqrt, Nat.floor, HMul.hMul, OfScientific.ofScientific, Bool.true, Green39.proportionCoverable, Filter.atTop, nhds, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_39.variant_101 : answer(sorry) ↔
    Tendsto
      (fun p : {q : ℕ // q.Prime} ↦
        let k := Nat.sqrt p
…
```
theorem Green39.green_39.variant_theta Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$ sorry line 120 source
Source documentation
Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$. [Gr24]
NOTE: using $C p^\theta$ translates as stated makes the conjecture trivially false by the pigeonhole principle. Indeed for a set of size $p^\theta$, we cover at most $C p^{2\theta}$ elements, which is strictly less than $p$ for $\theta < 1/2$. We interpret the question as asking whether $O(p^{1-\theta})$ translates suffice. This generalizes the main conjecture where $\sqrt{p} = p^{1-1/2}$.

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, LE.le, HDiv.hDiv, Nat, Exists, And, GT.gt, Filter.Tendsto, Subtype, Nat.Prime, Nat.floor, HPow.hPow, HMul.hMul, HSub.hSub, Green39.proportionCoverable, Filter.atTop and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
70 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_39.variant_theta : answer(sorry) ↔
    ∀ (θ : ℝ), 0 < θ → θ ≤ 1/2 →
    ∃ C > 1, Tendsto
      (fun p : {q : ℕ // q.Prime} ↦
        let k := ⌊(p : ℝ) ^ θ⌋₊
…
```