92243.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

11 declarations, 5 contain sorry, 0 declared as axioms

Tug of war score between prime gap increases and decreases

Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases: start with score $= 0$ at $n = 1$ and at stage $k > 1$, increase (resp. decrease) the score by $1$ if the $k$-th prime gap is greater (resp. less) than the previous prime gap.

References:

A092243

Imports (1)

FormalConjecturesUtil

def OeisA92243.primeGap The $n$-th prime gap $g(n) = p_n - p_{n-1}$ for $n \ge 1$, where $p_i$ is the $i$-th prime (0-indexed) line 36 source
Source documentation
The $n$-th prime gap $g(n) = p_n - p_{n-1}$ for $n \ge 1$, where $p_i$ is the $i$-th prime (0-indexed).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HSub.hSub, Nat.nth, Nat.Prime, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
noncomputable def primeGap (n : ℕ) : ℕ :=
…
```
def OeisA92243.a Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases line 40 source
Source documentation
Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: ite, LE.le, OfNat.ofNat, Finset.sum, Finset.Icc, Int.sign, HSub.hSub, OeisA92243.primeGap
12 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
theorem OeisA92243.a_0 Value of the sequence a at 0 line 48 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA92243.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by rfl
…
```
Mentions: OeisA92243.a
theorem OeisA92243.a_1 Value of the sequence a at 1 line 52 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA92243.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by rfl
…
```
Mentions: OeisA92243.a
theorem OeisA92243.a_2 Value of the sequence a at 2 line 56 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA92243.a, OfNat.ofNat, Nat
Proof uses: id, Not, LE.le, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, ite, Finset.sum, Finset.Icc, Int.sign, HSub.hSub, OeisA92243.primeGap, congrArg, if_neg, Finset, Singleton.singleton, Finset.sum_singleton and 5 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by
…
```
Mentions: OeisA92243.a
theorem OeisA92243.a_3 Value of the sequence a at 3 line 71 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA92243.a, OfNat.ofNat, Nat
Proof uses: id, Not, LE.le, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, ite, Finset.sum, Finset.Icc, Int.sign, HSub.hSub, OeisA92243.primeGap, congrArg, if_neg, Finset, Insert.insert, Singleton.singleton and 9 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by
…
```
Mentions: OeisA92243.a
theorem OeisA92243.conjecture1 Is the score $a(n) > 0$ for some $n > 250000$? sorry line 88 source
Source documentation
Is the score $a(n) > 0$ for some $n > 250000$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GT.gt, OfNat.ofNat, Int, OeisA92243.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 : answer(sorry) ↔ ∃ n > 250000, a n > 0 := by
…
```
Mentions: OeisA92243.a
theorem OeisA92243.conjecture2 Is the score $a(n)$ bounded from below? sorry line 94 source
Source documentation
Is the score $a(n)$ bounded from below?

depends on sorry what this means
Statement uses: Iff, True, Exists, Int, Nat, LE.le, OeisA92243.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 : answer(sorry) ↔ ∃ B : ℤ, ∀ n : ℕ, B ≤ a n := by
…
```
Mentions: OeisA92243.a
theorem OeisA92243.conjecture3 Is the score $a(n)$ bounded from above? sorry line 100 source
Source documentation
Is the score $a(n)$ bounded from above?

depends on sorry what this means
Statement uses: Iff, True, Exists, Int, Nat, LE.le, OeisA92243.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 : answer(sorry) ↔ ∃ B : ℤ, ∀ n : ℕ, a n ≤ B := by
…
```
Mentions: OeisA92243.a
theorem OeisA92243.conjecture4 Is the score $a(n) > 0$ for infinitely many values of $n$? sorry line 106 source
Source documentation
Is the score $a(n) > 0$ for infinitely many values of $n$?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, GT.gt, Int, OeisA92243.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture4 : answer(sorry) ↔ Set.Infinite {n : ℕ | a n > 0} := by
…
```
Mentions: OeisA92243.a
theorem OeisA92243.conjecture5 Is the score $a(n) < 0$ for infinitely many values of $n$? sorry line 112 source
Source documentation
Is the score $a(n) < 0$ for infinitely many values of $n$?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, LT.lt, Int, OeisA92243.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture5 : answer(sorry) ↔ Set.Infinite {n : ℕ | a n < 0} := by
…
```
Mentions: OeisA92243.a