49473.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

12 declarations, 1 contain sorry, 0 declared as axioms

Nearest integer to $n/\sqrt{2}$

Nearest integer to $n/\sqrt{2}$, defined by $\lfloor n/\sqrt{2} + 1/2 \rfloor$.

References:

A049473

Imports (1)

FormalConjecturesUtil

def OeisA49473.a Nearest integer to $n/\sqrt{2}$ line 33 source
Source documentation
Nearest integer to $n/\sqrt{2}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.toNat, Int.floor, Real, HAdd.hAdd, HDiv.hDiv, Real.sqrt, OfNat.ofNat
14 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA49473.a_0 line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA49473.a, OfNat.ofNat
Proof uses: id, Real, HAdd.hAdd, HDiv.hDiv, Real.sqrt, of_eq_true, Eq.trans, Inv.inv, True, congr, congrArg, congrFun', CharP.cast_eq_zero, zero_div, one_div, zero_add, eq_self, Eq.mpr, Int.toNat, Int.floor and 25 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
theorem OeisA49473.a_1 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA49473.a, OfNat.ofNat
Proof uses: id, Real, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Nat.cast_one, Eq.mpr, Int.toNat, Int.floor, HAdd.hAdd, HDiv.hDiv, Real.sqrt, congrArg, LT.lt, Iff.mpr, Real.sqrt_pos, Mathlib.Meta.NormNum.isNat_lt_true, Nat.cast_zero, Mathlib.Meta.NormNum.instAtLeastTwo and 103 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
theorem OeisA49473.a_2 line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA49473.a, OfNat.ofNat
Proof uses: id, Real, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Eq.mpr, Int.toNat, Int.floor, HAdd.hAdd, HDiv.hDiv, Real.sqrt, congrArg, LT.lt, Iff.mpr, Real.sqrt_pos, Mathlib.Meta.NormNum.isNat_lt_true, Nat.cast_zero, Bool and 103 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
theorem OeisA49473.a_3 line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA49473.a, OfNat.ofNat
Proof uses: id, Real, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Eq.mpr, Int.toNat, Int.floor, HAdd.hAdd, HDiv.hDiv, Real.sqrt, congrArg, LT.lt, Iff.mpr, Real.sqrt_pos, Mathlib.Meta.NormNum.isNat_lt_true, Nat.cast_zero, Bool and 104 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
theorem OeisA49473.a_4 line 121 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA49473.a, OfNat.ofNat
Proof uses: id, Real, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Eq.mpr, Int.toNat, Int.floor, HAdd.hAdd, HDiv.hDiv, Real.sqrt, congrArg, LT.lt, Iff.mpr, Real.sqrt_pos, Mathlib.Meta.NormNum.isNat_lt_true, Nat.cast_zero, Bool and 101 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
theorem OeisA49473.a_5 line 145 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA49473.a, OfNat.ofNat
Proof uses: id, Real, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Eq.mpr, Int.toNat, Int.floor, HAdd.hAdd, HDiv.hDiv, Real.sqrt, congrArg, LT.lt, Iff.mpr, Real.sqrt_pos, Mathlib.Meta.NormNum.isNat_lt_true, Nat.cast_zero, Bool and 101 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
def OeisA49473.zetaThreeReal $\zeta(3)$ (Apéry's constant) line 169 source
Source documentation
$\zeta(3)$ (Apéry's constant).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Complex.re, riemannZeta, OfNat.ofNat, Complex
3 instances, coercions and auxiliary constants left out.
```
noncomputable def zetaThreeReal : ℝ :=
…
```
def OeisA49473.s Let $s(n) = \zeta(3) - \sum_{k=1}^{n} 1/k^3$ line 173 source
Source documentation
Let $s(n) = \zeta(3) - \sum_{k=1}^{n} 1/k^3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HSub.hSub, OeisA49473.zetaThreeReal, Finset.sum, Finset.range, HDiv.hDiv, OfNat.ofNat, HPow.hPow, HAdd.hAdd
17 instances, coercions and auxiliary constants left out.
```
noncomputable def s (n : ℕ) : ℝ :=
…
```
def OeisA49473.A001953 A001953: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)\sqrt{2} \rfloor$ for $k \ge 0$ line 177 source
Source documentation
A001953: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)\sqrt{2} \rfloor$ for $k \ge 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Eq, Int.toNat, Int.floor, Real, HMul.hMul, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, Real.sqrt
16 instances, coercions and auxiliary constants left out.
```
def A001953 : Set ℕ :=
…
```
def OeisA49473.A001954 A001954: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)(2 + \sqrt{2}) \rfloor$ for $k \ge 0$ line 181 source
Source documentation
A001954: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)(2 + \sqrt{2}) \rfloor$ for $k \ge 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Eq, Int.toNat, Int.floor, Real, HMul.hMul, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, Real.sqrt
16 instances, coercions and auxiliary constants left out.
```
def A001954 : Set ℕ :=
…
```
theorem OeisA49473.conjecture sorry line 194 source
depends on sorry what this means
Statement uses: And, Nat, LE.le, OfNat.ofNat, LT.lt, Real, OeisA49473.s, OeisA49473.a, HDiv.hDiv, HPow.hPow, HSub.hSub, Iff, Eq, Membership.mem, Set, OeisA49473.A001954, OeisA49473.A001953
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.