341254.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Bounds on fractional parts $n r^2 - a(n)$ for $r = (2+\sqrt{5})/2$

$a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$.

References:

A341254
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA341254.rConst The constant $r = (2 + \sqrt{5})/2$ line 38 source
Source documentation
The constant $r = (2 + \sqrt{5})/2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: HDiv.hDiv, HAdd.hAdd, OfNat.ofNat, Real.sqrt
9 instances, coercions and auxiliary constants left out.
```
noncomputable def rConst : ℝ := (2 + sqrt 5) / 2
```
def OeisA341254.rSq The constant $r^2$ line 41 source
Source documentation
The constant $r^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: HMul.hMul, OeisA341254.rConst
2 instances, coercions and auxiliary constants left out.
```
noncomputable def rSq : ℝ := rConst * rConst
```
Mentions: OeisA341254.rConst
def OeisA341254.a $a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$ line 47 source
Source documentation
$a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$. Note: The original OEIS definition has $n$ starting at 1. We define $a(n)$ for all $\mathbb{N}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Real, OeisA341254.rConst, Int, Int.floor, HMul.hMul, Int.toNat
9 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA341254.a_1 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA341254.a, OfNat.ofNat
Proof uses: id, Real, OeisA341254.rConst, Int, Int.floor, HMul.hMul, Int.toNat, HDiv.hDiv, HAdd.hAdd, Real.sqrt, Eq.mpr, congrFun', congrArg, Nat.cast_one, mul_one, And, LE.le, LT.lt, propext, Int.floor_eq_iff and 103 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
lemma OeisA341254.a_2 line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA341254.a, OfNat.ofNat
Proof uses: id, Real, OeisA341254.rConst, Int, Int.floor, HMul.hMul, Int.toNat, HDiv.hDiv, HAdd.hAdd, Real.sqrt, Eq.mpr, And, LE.le, LT.lt, congrArg, propext, Int.floor_eq_iff, of_eq_true, Eq.trans, True and 102 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma OeisA341254.a_3 line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA341254.a, OfNat.ofNat
Proof uses: id, Real, OeisA341254.rConst, Int, Int.floor, HMul.hMul, Int.toNat, HDiv.hDiv, HAdd.hAdd, Real.sqrt, Eq.mpr, And, LE.le, LT.lt, congrArg, propext, Int.floor_eq_iff, of_eq_true, Eq.trans, True and 102 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma OeisA341254.a_4 line 115 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA341254.a, OfNat.ofNat
Proof uses: id, Real, OeisA341254.rConst, Int, Int.floor, HMul.hMul, Int.toNat, HDiv.hDiv, HAdd.hAdd, Real.sqrt, Eq.mpr, And, LE.le, LT.lt, congrArg, propext, Int.floor_eq_iff, of_eq_true, Eq.trans, True and 102 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma OeisA341254.a_5 line 135 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA341254.a, OfNat.ofNat
Proof uses: id, Real, OeisA341254.rConst, Int, Int.floor, HMul.hMul, Int.toNat, HDiv.hDiv, HAdd.hAdd, Real.sqrt, Eq.mpr, And, LE.le, LT.lt, congrArg, propext, Int.floor_eq_iff, of_eq_true, Eq.trans, True and 102 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
theorem OeisA341254.a_bounds sorry line 163 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, LT.lt, Real, HDiv.hDiv, HSub.hSub, HMul.hMul, OeisA341254.rSq, OeisA341254.a
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.