289411.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Symmetry of digit sum differences $\operatorname{sign}(S_5(k) - S_1(k))$

a: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$. $\mathrm{A007953}(n)$ is the digital sum of $n$ in base 10. The sequence is non-negative, so the sum over $\mathbb{Z}$ is converted to $\mathbb{N}$.

References:

A289411
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
A007953

Imports (1)

FormalConjecturesUtil

def OeisA289411.a a: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$ line 46 source
Source documentation
a: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$. $\mathrm{A007953}(n)$ is the digital sum of $n$ in base 10. The sequence is non-negative, so the sum over $\mathbb{Z}$ is converted to $\mathbb{N}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List.sum, Nat.digits, OfNat.ofNat, Int.toNat, Finset.sum, Int, Finset.range, HAdd.hAdd, Int.sign, HSub.hSub, HMul.hMul
12 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
lemma OeisA289411.a_0 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA289411.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA289411.a_1 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA289411.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA289411.a_2 line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA289411.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA289411.a_3 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA289411.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA289411.a_4 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA289411.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA289411.a_symm sorry line 78 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HSub.hSub, HDiv.hDiv, HPow.hPow, LE.le, Eq, OeisA289411.a, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

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