120424.lean`

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

9 declarations, 2 contain sorry, 0 declared as axioms

Half-Fibonacci sequence

$a(0) = 1, a(1) = 3$; for $n \ge 2$, $a(n) = f(a(n-1)) + f(a(n-2))$ where $f(x) = x/2$ if $x$ is even and $f(x) = x$ if $x$ is odd.

References:

A120424

Imports (1)

FormalConjecturesUtil

def OeisA120424.f Halve even numbers, leave odd numbers unchanged line 35 source
Source documentation
Halve even numbers, leave odd numbers unchanged.

no axioms what this means
Statement uses: Nat
Proof uses: ite, Eq, HMod.hMod, OfNat.ofNat, HDiv.hDiv
6 instances, coercions and auxiliary constants left out.
```
def f (x : ℕ) : ℕ := if x % 2 = 0 then x / 2 else x
```
def OeisA120424.a Half-Fibonacci sequence starting with $1, 3$ line 38 source
Source documentation
Half-Fibonacci sequence starting with $1, 3$.

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 1
  | 1 => 3
  | n + 2 => f (a (n + 1)) + f (a n)

/-- Value of the sequence `a` at 0. -/
@[category test, AMS 11]
```
Mentions: OeisA120424.f, test, AMS
theorem OeisA120424.a_0 Value of the sequence a at 0 line 45 source
Source documentation
Value of the sequence a at 0.

no axioms what this means
Statement uses: Eq, Nat, OeisA120424.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 = 1 := by rfl
…
```
Mentions: OeisA120424.a
theorem OeisA120424.a_1 Value of the sequence a at 1 line 49 source
Source documentation
Value of the sequence a at 1.

no axioms what this means
Statement uses: Eq, Nat, OeisA120424.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_1 : a 1 = 3 := by rfl
…
```
Mentions: OeisA120424.a
theorem OeisA120424.a_2 Value of the sequence a at 2 line 53 source
Source documentation
Value of the sequence a at 2.

no axioms what this means
Statement uses: Eq, Nat, OeisA120424.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_2 : a 2 = 4 := by rfl
…
```
Mentions: OeisA120424.a
theorem OeisA120424.a_3 Value of the sequence a at 3 line 57 source
Source documentation
Value of the sequence a at 3.

no axioms what this means
Statement uses: Eq, Nat, OeisA120424.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_3 : a 3 = 5 := by rfl
…
```
Mentions: OeisA120424.a
theorem OeisA120424.a_4 Value of the sequence a at 4 line 61 source
Source documentation
Value of the sequence a at 4.

no axioms what this means
Statement uses: Eq, Nat, OeisA120424.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_4 : a 4 = 7 := by rfl
…
```
Mentions: OeisA120424.a
theorem OeisA120424.conjecture1 Conjecture (1): The natural density of even terms in the sequence is $1/2$ sorry line 66 source
Source documentation
Conjecture (1): The natural density of even terms in the sequence is $1/2$.

depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Finset.card, Finset.filter, Eq, HMod.hMod, OeisA120424.a, OfNat.ofNat, Finset.range, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 :
    Filter.Tendsto (fun n : ℕ => ((Finset.filter (fun k => a k % 2 = 0) (Finset.range n)).card
        : ℝ) / (n : ℝ))
      Filter.atTop (nhds (1 / 2 : ℝ)) := by
…
```
Mentions: OeisA120424.a
theorem OeisA120424.conjecture2 Conjecture (2): There are infinitely many consecutive pairs that differ by 1 sorry line 75 source
Source documentation
Conjecture (2): There are infinitely many consecutive pairs that differ by 1.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, Or, Eq, OeisA120424.a, HAdd.hAdd, OfNat.ofNat
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 conjecture2 :
    Set.Infinite {n : ℕ | a (n + 1) = a n + 1 ∨ a n = a (n + 1) + 1} := by
…
```
Mentions: OeisA120424.a