167918.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

26 declarations, 3 contain sorry, 0 declared as axioms

Smallest index $k > n$ such that $(p_k+p_{k+1})/(p_n+p_{n+1})$ is an integer $\ge 2$

References:

A167918

Imports (1)

FormalConjecturesUtil

def OeisA167918.P $P(i)$ is the $i$-th prime, 1-indexed line 31 source
Source documentation
$P(i)$ is the $i$-th prime, 1-indexed.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Nat.Prime, HSub.hSub, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
noncomputable def P (i : ℕ) : ℕ := Nat.nth Nat.Prime (i - 1)
```
def OeisA167918.S $S(i) = p_i + p_{i+1}$ line 34 source
Source documentation
$S(i) = p_i + p_{i+1}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, OeisA167918.P, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
noncomputable def S (i : ℕ) : ℕ := P i + P (i + 1)
```
Mentions: OeisA167918.P
def OeisA167918.a Smallest index $k > n$ such that $S(n) \mid S(k)$ line 37 source
Source documentation
Smallest index $k > n$ such that $S(n) \mid S(k)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, InfSet.sInf, Set.ofPred, And, GT.gt, Dvd.dvd, OeisA167918.S
5 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA167918.a_0 Value of the sequence a at 0 line 43 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.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: OeisA167918.a
lemma OeisA167918.P_1 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: Nat.nth_prime_zero_eq_two
1 instances, coercions and auxiliary constants left out.
lemma OeisA167918.P_2 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: Nat.nth_prime_one_eq_three
1 instances, coercions and auxiliary constants left out.
lemma OeisA167918.P_3 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: Nat.nth_prime_two_eq_five
1 instances, coercions and auxiliary constants left out.
lemma OeisA167918.P_4 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: Nat.nth_prime_three_eq_seven
1 instances, coercions and auxiliary constants left out.
lemma OeisA167918.P_5 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: Nat.nth_prime_four_eq_eleven
1 instances, coercions and auxiliary constants left out.
lemma OeisA167918.P_6 line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.P_7 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.P_8 line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.P, OfNat.ofNat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_1 line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_1, OeisA167918.P_2, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_2 line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_2, OeisA167918.P_3, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_3 line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_3, OeisA167918.P_4, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_4 line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_4, OeisA167918.P_5, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_5 line 91 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_5, OeisA167918.P_6, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_6 line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_6, OeisA167918.P_7, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA167918.S_7 line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.S, OfNat.ofNat
Proof uses: id, Eq.mpr, HAdd.hAdd, OeisA167918.P, congrArg, OeisA167918.P_7, OeisA167918.P_8, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA167918.a_1 Value of the sequence a at 1 line 101 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, GT.gt, Dvd.dvd, OeisA167918.S, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, congrArg, OeisA167918.S_1, OeisA167918.S_6 and 29 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 40 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 6 := by
…
```
Mentions: OeisA167918.a
theorem OeisA167918.a_2 Value of the sequence a at 2 line 131 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, GT.gt, Dvd.dvd, OeisA167918.S, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, congrArg, OeisA167918.S_2, OeisA167918.S_5 and 27 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 5 := by
…
```
Mentions: OeisA167918.a
theorem OeisA167918.a_3 Value of the sequence a at 3 line 155 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, GT.gt, Dvd.dvd, OeisA167918.S, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, congrArg, OeisA167918.S_3, OeisA167918.S_5 and 25 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 5 := by
…
```
Mentions: OeisA167918.a
theorem OeisA167918.a_4 Value of the sequence a at 4 line 176 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA167918.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, GT.gt, Dvd.dvd, OeisA167918.S, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, congrArg, OeisA167918.S_4, OeisA167918.S_7 and 27 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 7 := by
…
```
Mentions: OeisA167918.a
theorem OeisA167918.conjecture_infinite Conjecture: the sequence is infinite, that is, for every $n \geq 1$ there is some $k > n$ with $S(n) \mid S(k)$, so that $a(n)$ is defined sorry line 203 source
Source documentation
Conjecture: the sequence is infinite, that is, for every $n \geq 1$ there is some $k > n$ with $S(n) \mid S(k)$, so that $a(n)$ is defined.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Exists, And, Dvd.dvd, OeisA167918.S
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture_infinite (n : ℕ) (hn : n > 0) : ∃ k > n, S n ∣ S k := by
…
```
Mentions: OeisA167918.S
theorem OeisA167918.conjecture1 Conjecture: $f(n, k) = 2$ for infinitely many cases, where $k = a(n)$ sorry line 213 source
Source documentation
Conjecture: $f(n, k) = 2$ for infinitely many cases, where $k = a(n)$.
No hypothesis on the existence of $a(n)$ is needed: if no suitable $k > n$ exists then $a(n) = 0$ and $S(0) = 4 < 2 S(n)$, so such $n$ cannot be witnesses.

depends on sorry what this means
Statement uses: Nat, Exists, And, GE.ge, GT.gt, OfNat.ofNat, Eq, OeisA167918.S, OeisA167918.a, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 (M : ℕ) :
    ∃ n : ℕ, n ≥ M ∧ n > 0 ∧ S (a n) = 2 * S n := by
…
```
Mentions: OeisA167918.S, OeisA167918.a
theorem OeisA167918.conjecture2 Open problem: is the ratio $f(n, k)$ bounded, where $k = a(n)$? sorry line 225 source
Source documentation
Open problem: is the ratio $f(n, k)$ bounded, where $k = a(n)$?
No hypothesis on the existence of $a(n)$ is needed: if no suitable $k > n$ exists then $a(n) = 0$ and $S(0) / S(n) = 4 / S(n) = 0$ for $n > 0$, so such $n$ do not affect boundedness. When $a(n) \neq 0$ we have $S(n) \mid S(a(n))$ by definition, so the division is exact.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, GT.gt, OfNat.ofNat, LE.le, HDiv.hDiv, OeisA167918.S, OeisA167918.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 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) ↔ ∃ C : ℕ, ∀ n : ℕ, n > 0 → S (a n) / S n ≤ C := by
…
```
Mentions: OeisA167918.S, OeisA167918.a