317.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 317

Reference: erdosproblems.com/317

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

theorem Erdos317.erdos_317 Is there some constant sorry line 37 source
Source documentation
Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $$0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$$

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, GE.ge, Fin, Rat, LE.le, Set, Set.range, Insert.insert, Neg.neg, Singleton.singleton, LT.lt, abs, Finset.sum and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
66 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_317 : answer(sorry) ↔
    ∃ c > 0, ∀ n ≥ 1, ∃ δ : Fin n → ℚ,
      Set.range δ ⊆ {-1, 0, 1} ∧
      letI lhs : ℝ := |∑ k, (δ k) / (k + 1)|
…
```
Mentions: c
theorem Erdos317.erdos_317.variants.claim2 Is it true that for sufficiently large sorry line 50 source
Source documentation
Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, $$\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$$ whenever the left-hand side is not zero?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Fin, Rat, LE.le, Set, Set.image, Set.univ, Insert.insert, Neg.neg, OfNat.ofNat, Singleton.singleton, Ne, abs, Finset.sum, Finset.univ, HDiv.hDiv, HAdd.hAdd and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_317.variants.claim2 : answer(sorry) ↔
    ∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} →
    letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))|
…
```
lemma Erdos317.claim2_inequality Inequality in erdos_317.variants.claim2 is obvious, the problem is strict inequality sorry line 60 source
Source documentation
Inequality in erdos_317.variants.claim2 is obvious, the problem is strict inequality.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Fin, Rat, LE.le, Set, Set.image, Set.univ, Insert.insert, Neg.neg, OfNat.ofNat, Singleton.singleton, Ne, abs, Finset.sum, Finset.univ, HDiv.hDiv, HAdd.hAdd, GE.ge, Finset.lcm and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma claim2_inequality : ∀ᶠ n in atTop,
    ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} →
    letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))|
…
```
theorem Erdos317.erdos_317.variants.counterexample erdos_317.variants.claim2 fails for small $n$, for example $$\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$$ line 71 source
Source documentation
erdos_317.variants.claim2 fails for small $n$, for example $$\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Fin, OfNat.ofNat, Nat, Rat, LE.le, Set, Set.image, Set.univ, Insert.insert, Neg.neg, Singleton.singleton, Ne, abs, Finset.sum, Finset.univ, HDiv.hDiv, HAdd.hAdd, GT.gt, Finset.lcm and 2 more
Proof uses: Eq.mpr, Exists, And, Eq, Eq.trans, Mathlib.Tactic.Push.not_forall_eq, congrArg, funext, Exists.intro, Matrix.vecCons, Matrix.vecEmpty, Inv.inv, congr, congrFun', Set.range, Set.image_univ, Nat.succ, Union.union, Matrix.range_cons, EmptyCollection.emptyCollection and 76 more
121 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_317.variants.counterexample : ¬ (∀  δ : (Fin 4) → ℚ, δ '' Set.univ ⊆ {-1,0,1} →
    letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))|
…
```