2897.lean`

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

10 declarations, 1 contain sorry, 0 declared as axioms

Coefficient of $(xyz)^n$ in $((x+y)(y+z)(z+x))^n$ equaling $\binom{2n}{n}^3$

The sequence $a(n)$ is defined by $a(n) = \binom{2n}{n}^3$.

References:

A002897
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA2897.a The sequence $a(n)$ is defined by $a(n) = \binom{2n}{n}^3$ line 43 source
Source documentation
The sequence $a(n)$ is defined by $a(n) = \binom{2n}{n}^3$. We use Nat.choose (2 * n) n for the central binomial coefficient.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: HPow.hPow, Nat.choose, HMul.hMul, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := (Nat.choose (2 * n) n) ^ 3
```
abbrev OeisA2897.Vars line 45 source
no axioms what this means
Proof uses: Fin, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.
def OeisA2897.xyzPowN The finsupp corresponding to the monomial $x^n y^n z^n$ line 51 source
Source documentation
The finsupp corresponding to the monomial $x^n y^n z^n$. This is the map $\lambda i. n$. Since Fin 3 is finite, this function is finitely supported.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finsupp, OeisA2897.Vars
Proof uses: Finsupp.ofSupportFinite
5 instances, coercions and auxiliary constants left out.
```
noncomputable def xyzPowN (n : ℕ) : Finsupp Vars ℕ :=
…
```
Mentions: OeisA2897.Vars
def OeisA2897.pPoly The polynomial $pPoly(X, Y, Z) = (1 + X + Y + Z)^{2n} (1 + X + Y - Z)^n (1 + X - Y + Z)^n$ line 60 source
Source documentation
The polynomial $pPoly(X, Y, Z) = (1 + X + Y + Z)^{2n} (1 + X + Y - Z)^n (1 + X - Y + Z)^n$. We identify $X_0, X_1, X_2$ with $X, Y, Z$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, MvPolynomial, OeisA2897.Vars, Int
Proof uses: MvPolynomial.X, OfNat.ofNat, HAdd.hAdd, Finsupp, HSub.hSub, HMul.hMul, HPow.hPow
33 instances, coercions and auxiliary constants left out.
```
noncomputable def pPoly (n : ℕ) : P :=
…
```
lemma OeisA2897.a_0 line 71 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2897.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 OeisA2897.a_1 line 74 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2897.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 OeisA2897.a_2 line 77 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2897.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 OeisA2897.a_3 line 80 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2897.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 OeisA2897.a_4 line 83 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2897.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 OeisA2897.a_eq_coeff sorry line 94 source
depends on sorry what this means
Statement uses: Nat, Eq, Int, OeisA2897.a, MvPolynomial.coeff, OeisA2897.Vars, OeisA2897.xyzPowN, OeisA2897.pPoly
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

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