108.lean`

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

11 declarations, 1 contain sorry, 0 declared as axioms

Fractional parts of sums of reciprocals of Catalan numbers $C(n)$

Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$.

The sum $\sum_{i=j}^k \frac{1}{a(i)}$ of reciprocals of Catalan numbers.

References:

A000108
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA108.a Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$ line 42 source
Source documentation
Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$.

no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, Nat.choose, HMul.hMul, OfNat.ofNat, HAdd.hAdd
7 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := (Nat.choose (2 * n) n) / (n + 1)
```
def OeisA108.aRat line 44 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: Inv.inv, OeisA108.a
3 instances, coercions and auxiliary constants left out.
def OeisA108.catalanReciprocalSum The sum $\sum_{i=j}^k \frac{1}{a(i)}$ of reciprocals of Catalan numbers line 47 source
Source documentation
The sum $\sum_{i=j}^k \frac{1}{a(i)}$ of reciprocals of Catalan numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: Finset.sum, Finset.Icc, OeisA108.aRat
3 instances, coercions and auxiliary constants left out.
```
def catalanReciprocalSum (j k : ℕ) : ℚ :=
…
```
def OeisA108.IndexCond The index condition on $(j, k)$ from the conjecture: $0 < \min\{2,k\} \le j \le k$ line 52 source
Source documentation
The index condition on $(j, k)$ from the conjecture: $0 < \min\{2,k\} \le j \le k$. Since j and k are natural numbers, $0 < \min\{2,k\}$ is equivalent to $1 \le k$.

no axioms what this means
Statement uses: Nat
Proof uses: And, LE.le, OfNat.ofNat, Min.min
3 instances, coercions and auxiliary constants left out.
```
def IndexCond (j k : ℕ) : Prop :=
…
```
def OeisA108.fracPart The fractional part of a rational number line 58 source
Source documentation
The fractional part of a rational number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat
Proof uses: Int.fract
4 instances, coercions and auxiliary constants left out.
```
def fracPart (q : ℚ) : ℚ := fract q
…
```
lemma OeisA108.a_0 line 62 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108.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.
lemma OeisA108.a_1 line 65 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108.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.
lemma OeisA108.a_2 line 68 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108.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.
lemma OeisA108.a_3 line 71 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108.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.
lemma OeisA108.a_4 line 74 source
no axioms what this means
Statement uses: Eq, Nat, OeisA108.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 OeisA108.catalanReciprocalSum_fracPart_inj sorry line 86 source
depends on sorry what this means
Statement uses: Nat, OeisA108.IndexCond, Ne, Prod, Prod.mk, Rat, OeisA108.fracPart, OeisA108.catalanReciprocalSum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

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