36.lean`

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

4 declarations, 2 contain sorry, 0 declared as axioms

Green's Open Problem 36

References:

[Gr24] Green's Open Problems #36
[CKS05] Cohn, H., Kleinberg, R., Szegedy, B., and Umans, C. "Group-theoretic Algorithms for Matrix Multiplication" (Problem 4.7)

Imports (1)

FormalConjecturesUtil

def Green36.SimultaneousDoubleProduct The simultaneous double product property [CKS05, 4.1] line 37 source
Source documentation
The simultaneous double product property [CKS05, 4.1].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finset
Proof uses: And, Eq, Nat, Finset.card, HAdd.hAdd, Classical.propDecidable, HMul.hMul, Ne, Disjoint
10 instances, coercions and auxiliary constants left out.
```
def SimultaneousDoubleProduct {ι H : Type*} [AddCommGroup H]
    (A B : ι → Finset H) : Prop :=
…
```
def Green36.Green36Property A variant of the simultaneous double product property, as stated in [Gr24, Problem 36] line 44 source
Source documentation
A variant of the simultaneous double product property, as stated in [Gr24, Problem 36].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finset
Proof uses: And, Eq, Nat, Finset.card, HAdd.hAdd, Classical.propDecidable, HMul.hMul, Ne, Disjoint
10 instances, coercions and auxiliary constants left out.
```
def Green36Property {ι H : Type*} [AddCommGroup H]
    (A B : ι → Finset H) : Prop :=
…
```
theorem Green36.green_36 Do the following exist, for arbitrarily large $n$? sorry line 60 source
Source documentation
Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$, together with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \ge n^{2-o(1)}$ and $|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$ ($j \neq k$)?
NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for $i \neq k$. See green_36.variants.cks05.

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, Exists, AddCommGroup, Finite, Fin, Finset, And, LE.le, HPow.hPow, HSub.hSub, Nat.card, HAdd.hAdd, HMul.hMul, Finset.card 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 green_36 :
    answer(sorry) ↔
      ∀ ε > (0 : ℝ), ∃ᶠ n in atTop,
        ∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),
          (n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧
          (∀ i, (n : ℝ) ^ (2 - ε) ≤ (A i).card * (B i).card) ∧
          Green36Property A B := by
…
```
Mentions: Green36.Green36Property
theorem Green36.green_36.variants.cks05 Variant using the exact simultaneous double product property from [CKS05, 4.1] sorry line 71 source
Source documentation
Variant using the exact simultaneous double product property from [CKS05, 4.1].

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, Exists, AddCommGroup, Finite, Fin, Finset, And, LE.le, HPow.hPow, HSub.hSub, Nat.card, HAdd.hAdd, HMul.hMul, Finset.card 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 green_36.variants.cks05 :
    answer(sorry) ↔
      ∀ ε > (0 : ℝ), ∃ᶠ n in atTop,
        ∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),
          (n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧
          (∀ i, (n : ℝ) ^ (2 - ε) ≤ (A i).card * (B i).card) ∧
          SimultaneousDoubleProduct A B := by
…
```
Mentions: Green36.SimultaneousDoubleProduct