1188.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1188

Reference: erdosproblems.com/1188

Imports (1)

FormalConjecturesUtil

abbrev Erdos1188.CongruenceClass A congruence class (n, a) represents a (mod n) line 34 source
Source documentation
A congruence class (n, a) represents a (mod n).

no axioms what this means
Proof uses: Prod, Nat
```
abbrev CongruenceClass := ℕ × ℕ
```
def Erdos1188.ValidClass The modulus is > 1 and the residue is reduced line 37 source
Source documentation
The modulus is > 1 and the residue is reduced.

no axioms what this means
Statement uses: Erdos1188.CongruenceClass
Proof uses: And, LE.le, Nat, OfNat.ofNat, Prod.fst, LT.lt, Prod.snd
3 instances, coercions and auxiliary constants left out.
```
def ValidClass (c : CongruenceClass) : Prop := 2 ≤ c.1 ∧ c.2 < c.1
```
Mentions: c, Erdos1188.CongruenceClass
def Erdos1188.Satisfies z lies in the class (n, a) line 40 source
Source documentation
z lies in the class (n, a).

no axioms what this means
Statement uses: Int, Erdos1188.CongruenceClass
Proof uses: Eq, HMod.hMod, Prod.fst, Nat, Prod.snd
4 instances, coercions and auxiliary constants left out.
```
def Satisfies (z : ℤ) (c : CongruenceClass) : Prop := z % (c.1 : ℤ) = (c.2 : ℤ)
```
Mentions: c, Erdos1188.CongruenceClass
def Erdos1188.Covers Every integer satisfies some congruence of S line 43 source
Source documentation
Every integer satisfies some congruence of S.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos1188.CongruenceClass
Proof uses: Int, Exists, And, Membership.mem, Erdos1188.Satisfies
2 instances, coercions and auxiliary constants left out.
```
def Covers (S : Finset CongruenceClass) : Prop := ∀ z : ℤ, ∃ c ∈ S, Satisfies z c
```
Mentions: Erdos1188.CongruenceClass, c, Erdos1188.Satisfies
def Erdos1188.HasDistinctModuli No two classes of S share a modulus line 46 source
Source documentation
No two classes of S share a modulus.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos1188.CongruenceClass
Proof uses: Membership.mem, Eq, Nat, Prod.fst
2 instances, coercions and auxiliary constants left out.
```
def HasDistinctModuli (S : Finset CongruenceClass) : Prop :=
…
```
Mentions: Erdos1188.CongruenceClass
def Erdos1188.IsMinimalDistinctCoveringSystem A minimal distinct covering system: valid classes with distinct moduli that cover ℤ, with no proper subset already covering line 51 source
Source documentation
A minimal distinct covering system: valid classes with distinct moduli that cover ℤ, with no proper subset already covering.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos1188.CongruenceClass
Proof uses: And, Membership.mem, Erdos1188.ValidClass, Erdos1188.HasDistinctModuli, Erdos1188.Covers, LT.lt, Not
5 instances, coercions and auxiliary constants left out.
```
def IsMinimalDistinctCoveringSystem (S : Finset CongruenceClass) : Prop :=
…
```
Mentions: Erdos1188.CongruenceClass
def Erdos1188.ClassesUpTo All classes (n, a) with 2 ≤ n ≤ x and a < n line 56 source
Source documentation
All classes (n, a) with 2 ≤ n ≤ x and a < n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Erdos1188.CongruenceClass
Proof uses: Finset.biUnion, Finset.Icc, OfNat.ofNat, Finset.image, Prod.mk, Finset.range
5 instances, coercions and auxiliary constants left out.
```
def ClassesUpTo (x : ℕ) : Finset CongruenceClass :=
…
```
Mentions: Erdos1188.CongruenceClass
def Erdos1188.coveringCount F(x): the number of minimal distinct covering systems all of whose moduli lie in [1, x] line 62 source
Source documentation
F(x): the number of minimal distinct covering systems all of whose moduli lie in [1, x].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset, Erdos1188.CongruenceClass, Finset.filter, Erdos1188.IsMinimalDistinctCoveringSystem, Classical.propDecidable, Finset.powerset, Erdos1188.ClassesUpTo
```
noncomputable def coveringCount (x : ℕ) : ℕ :=
…
```
theorem Erdos1188.erdos_1188 Call a set of distinct integers sorry line 75 source
Source documentation
Call a set of distinct integers $1<n_1<\cdots<n_k$ with associated congruence classes $a_i\pmod{n_i}$ a distinct covering system if every integer satisfies at least one of these congruences. A minimal distinct covering system is one such that no proper subset forms a covering system. Let $F(x)$ count the number of minimal distinct covering systems with all moduli in $[1,x]$. Estimate $F(x)$.
The estimate is log(log F(x)) / log x → 1.

depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Real.log, Erdos1188.coveringCount, Filter.atTop, nhds, OfNat.ofNat
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_1188 :
    Tendsto (fun x : ℕ => Real.log (Real.log (coveringCount x : ℝ)) / Real.log (x : ℝ))
      atTop (𝓝 1) := by
…
```
Mentions: Erdos1188.coveringCount