182510.lean`

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

10 declarations, 2 contain sorry, 0 declared as axioms

Recurrence with bitwise XOR

The sequence is defined by $a(0) = 0$, $a(1) = 1$, and for $n \ge 0$, $$a(n+2) = (a(n+1) \mathbin{\mathrm{XOR}} (n+2)) - a(n),$$ where $\mathrm{XOR}$ is the bitwise exclusive-or operator on integers.

References:

A182510

Imports (1)

FormalConjecturesUtil

def OeisA182510.a Defining recurrence for $a(n)$ line 35 source
Source documentation
Defining recurrence for $a(n)$.

standard axioms: propext what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℤ
  | 0 => 0
  | 1 => 1
  | n + 2 => Int.xor (a (n + 1)) (n + 2 : ℤ) - a n

@[category test, AMS 11]
```
Mentions: test, AMS
theorem OeisA182510.a_0 line 41 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.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 OeisA182510.a_1 line 44 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.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 OeisA182510.a_2 line 47 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.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 OeisA182510.a_3 line 50 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA182510.a_4 line 53 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA182510.a_5 line 56 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA182510.a_6 line 59 source
standard axioms: propext what this means
Statement uses: Eq, Int, OeisA182510.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 OeisA182510.conjecture1 Conjecture: the sequence contains 8 zeros sorry line 64 source
Source documentation
Conjecture: the sequence contains 8 zeros.

depends on sorry what this means
Statement uses: Eq, Nat, Set.ncard, Set.ofPred, Int, OeisA182510.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 : Set.ncard {n : ℕ | a n = 0} = 8 := by
…
```
Mentions: OeisA182510.a
theorem OeisA182510.conjecture2 Conjecture: more positive terms than negative sorry line 75 source
Source documentation
Conjecture: more positive terms than negative.
As $n \to \infty$, the count of positive terms is greater than the count of negative terms.
The requirement that $n$ is large enough is needed, since the claim fails for $n = 100$.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LT.lt, Finset.card, Finset.filter, Int, OeisA182510.a, OfNat.ofNat, Finset.range, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 :
    ∀ᶠ n in Filter.atTop,
      ((Finset.range n).filter (fun k => a k < 0)).card <
        ((Finset.range n).filter (fun k => 0 < a k)).card := by
…
```
Mentions: OeisA182510.a