211417.lean`

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

15 declarations, 7 contain sorry, 0 declared as axioms

Integrality and supercongruences of the factorial ratio $\frac{(6n)! n!}{(3n)! (2n)!^2}$

Integral factorial ratio sequence: $$a(n) = \frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$

References:

A211417
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
[Me11] R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 (2011).

Imports (1)

FormalConjecturesUtil

def OeisA211417.a Integral factorial ratio sequence: $$a(n) = \frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$ line 42 source
Source documentation
Integral factorial ratio sequence: $$a(n) = \frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$

no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HMul.hMul, Nat.factorial, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA211417.coprimeIndices line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.filter, And, LE.le, OfNat.ofNat, Eq, Nat.gcd, Finset.range, HAdd.hAdd
7 instances, coercions and auxiliary constants left out.
def OeisA211417.divisorProduct The product term in the denominator of the general conjecture line 56 source
Source documentation
The product term in the denominator of the general conjecture: $$\prod_{i = 1..r, i \text{ coprime to } 30} (30n - i)$$ We define this in ℤ to handle the $n=0$ case where $30n-i$ in the product might be negative.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Finset.prod, OeisA211417.coprimeIndices, HSub.hSub, HMul.hMul, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def divisorProduct (n r : ℕ) : ℤ :=
…
```
lemma OeisA211417.a_0 line 61 source
no axioms what this means
Statement uses: Eq, Nat, OeisA211417.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 OeisA211417.a_1 line 64 source
no axioms what this means
Statement uses: Eq, Nat, OeisA211417.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 OeisA211417.a_2 line 67 source
no axioms what this means
Statement uses: Eq, Nat, OeisA211417.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 OeisA211417.a_3 line 70 source
no axioms what this means
Statement uses: Eq, Nat, OeisA211417.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 OeisA211417.a_4 line 73 source
no axioms what this means
Statement uses: Eq, Nat, OeisA211417.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 OeisA211417.thirty_mul_sub_one_dvd_a sorry line 84 source
depends on sorry what this means
Statement uses: Nat, Dvd.dvd, Int, HSub.hSub, HMul.hMul, OfNat.ofNat, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA211417.seven_mul_a_dvd_two_mul_add_one sorry line 94 source
depends on sorry what this means
Statement uses: Nat, Dvd.dvd, Int, HAdd.hAdd, HMul.hMul, OfNat.ofNat, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA211417.a_dvd_three_mul_add_one sorry line 105 source
depends on sorry what this means
Statement uses: Nat, Dvd.dvd, Int, HAdd.hAdd, HMul.hMul, OfNat.ofNat, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA211417.a_dvd_five_mul_add_one sorry line 116 source
depends on sorry what this means
Statement uses: Nat, Dvd.dvd, Int, HAdd.hAdd, HMul.hMul, OfNat.ofNat, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA211417.forty_two_mul_a_dvd_product sorry line 127 source
depends on sorry what this means
Statement uses: Nat, Dvd.dvd, Int, HMul.hMul, HAdd.hAdd, OfNat.ofNat, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA211417.general_divisibility sorry line 151 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Int, And, LT.lt, Dvd.dvd, OeisA211417.divisorProduct, HMul.hMul, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA211417.supercongruence sorry line 173 source
depends on sorry what this means
Statement uses: Nat, Nat.Prime, LE.le, OfNat.ofNat, LT.lt, Dvd.dvd, Int, HPow.hPow, HMul.hMul, HSub.hSub, OeisA211417.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

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