340737.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Convergence of fraction numerators $a(n)/b(n)$ to $e$

a: Numerators of a sequence of fractions converging to $e$. $$a(1) = 3, a(2) = 5$$ For $n > 2$: $$a(n) = \begin{cases} \left(\frac{n+2}{2}\right) a(n-1) - a(n-2) - \left(\frac{n-2}{2}\right) a(n-3) & \text{if } n \text{ is even} \cr 2 a(n-1) + n a(n-2) & \text{if } n \text{ is odd} \end{cases}$$

References:

A340737
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
b

Imports (1)

FormalConjecturesUtil

def OeisA340737.a Numerators of a sequence of fractions converging to $e$ line 55 source
Source documentation
Numerators of a sequence of fractions converging to $e$. $$a(1) = 3, a(2) = 5$$ For $n > 2$: $$a(n) = \begin{cases} \left(\frac{n+2}{2}\right) a(n-1) - a(n-2) - \left(\frac{n-2}{2}\right) a(n-3) & \text{if } n \text{ is even} \cr 2 a(n-1) + n a(n-2) & \text{if } n \text{ is odd} \end{cases}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: WellFounded.Nat.fix, InvImage, LT.lt, OeisA340737.a.match_1, Unit, OfNat.ofNat, Nat.succ, HAdd.hAdd, HSub.hSub, dite, Eq, HMod.hMod, HDiv.hDiv, Int.toNat, Int, HMul.hMul, Int.ofNat, Not
18 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA340737.b Denominators of a sequence of fractions converging to $e$ line 90 source
Source documentation
Denominators of a sequence of fractions converging to $e$. This sequence is defined by the same recurrence relation as a but with initial values $b(1)=1, b(2)=2$. $$b(1) = 1, b(2) = 2$$ For $n > 2$: $$b(n) = \begin{cases} \left(\frac{n+2}{2}\right) b(n-1) - b(n-2) - \left(\frac{n-2}{2}\right) b(n-3) & \text{if } n \text{ is even} \cr 2 b(n-1) + n b(n-2) & \text{if } n \text{ is odd} \end{cases}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: WellFounded.Nat.fix, InvImage, LT.lt, OeisA340737.a.match_1, Unit, OfNat.ofNat, Nat.succ, HAdd.hAdd, HSub.hSub, dite, Eq, HMod.hMod, HDiv.hDiv, Int.toNat, Int, HMul.hMul, Int.ofNat, Not
18 instances, coercions and auxiliary constants left out.
```
def b (n : ℕ) : ℕ :=
…
```
lemma OeisA340737.a_1 line 117 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA340737.a, OfNat.ofNat
Proof uses: Eq.mpr, id, congrFun', congrArg, OeisA340737.a.eq_def, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA340737.a_2 line 120 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA340737.a, OfNat.ofNat
Proof uses: Eq.mpr, id, congrFun', congrArg, OeisA340737.a.eq_def, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA340737.a_3 line 123 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA340737.a, OfNat.ofNat
Proof uses: Eq.mpr, HAdd.hAdd, HSub.hSub, ite, HMod.hMod, HDiv.hDiv, Int.toNat, Int, HMul.hMul, Int.ofNat, id, congrFun', congrArg, OeisA340737.a.eq_def, OeisA340737.a.match_1, Unit, have_congr', Eq.refl, have_val_congr', have_body_congr' and 9 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA340737.a_4 line 126 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA340737.a, OfNat.ofNat
Proof uses: Eq.mpr, HAdd.hAdd, HSub.hSub, ite, HMod.hMod, HDiv.hDiv, Int.toNat, Int, HMul.hMul, Int.ofNat, id, congrFun', congrArg, OeisA340737.a.eq_def, OeisA340737.a.match_1, Unit, have_congr', Eq.refl, have_val_congr', have_body_congr' and 15 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA340737.a_5 line 129 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA340737.a, OfNat.ofNat
Proof uses: Eq.mpr, HAdd.hAdd, HSub.hSub, ite, HMod.hMod, HDiv.hDiv, Int.toNat, Int, HMul.hMul, Int.ofNat, id, congrFun', congrArg, OeisA340737.a.eq_def, OeisA340737.a.match_1, Unit, have_congr', Eq.refl, have_val_congr', have_body_congr' and 15 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA340737.tendsto_exp_one sorry line 140 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, OeisA340737.a, OeisA340737.b, Filter.atTop, nhds, Real.exp, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

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