40.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

7 declarations, 1 contain sorry, 0 declared as axioms

The prime numbers

The $n$-th prime number $p_n$, where $p_1 = 2$.

References:

A000040

Imports (1)

FormalConjecturesUtil

def OeisA40.a The sequence of prime numbers, with $a(0) = 0$ and $a(n) = p_n$ for $n \ge 1$ line 34 source
Source documentation
The sequence of prime numbers, with $a(0) = 0$ and $a(n) = p_n$ for $n \ge 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA40.a.match_1, Unit, OfNat.ofNat, Nat.nth, Nat.Prime
1 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA40.a_0 Value of the sequence a at 0 line 41 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA40.a, OfNat.ofNat
Proof uses: Eq.refl
2 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: OeisA40.a
theorem OeisA40.a_1 Value of the sequence a at 1 line 45 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA40.a, OfNat.ofNat
Proof uses: id, Nat.nth_prime_zero_eq_two
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by
…
```
Mentions: OeisA40.a
theorem OeisA40.a_2 Value of the sequence a at 2 line 51 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA40.a, OfNat.ofNat
Proof uses: id, Nat.nth_prime_one_eq_three
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 3 := by
…
```
Mentions: OeisA40.a
theorem OeisA40.a_3 Value of the sequence a at 3 line 57 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA40.a, OfNat.ofNat
Proof uses: id, Nat.nth_prime_two_eq_five
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 5 := by
…
```
Mentions: OeisA40.a
theorem OeisA40.a_4 Value of the sequence a at 4 line 63 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA40.a, OfNat.ofNat
Proof uses: id, Nat.nth_prime_three_eq_seven
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 7 := by
…
```
Mentions: OeisA40.a
theorem OeisA40.conjecture Conjecture from Thomas Ordowski (2023): $\log \log a(n+1) - \log \log a(n) < 1/n$ for $n > 0$ sorry line 72 source
Source documentation
Conjecture from Thomas Ordowski (2023): $\log \log a(n+1) - \log \log a(n) < 1/n$ for $n > 0$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Real, HSub.hSub, Real.log, OeisA40.a, HAdd.hAdd, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 0 < n) :
    Real.log (Real.log (a (n + 1) : ℝ)) - Real.log (Real.log (a n : ℝ)) < 1 / (n : ℝ) := by
…
```
Mentions: OeisA40.a