267581.lean`

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

11 declarations, 1 contain sorry, 0 declared as axioms

Recurrence for Rule 167 cellular automaton sequence

Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell.

References:

A267581
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA267581.caRule167 The rule function for Rule 167 line 39 source
Source documentation
The rule function for Rule 167. Inputs must be 0 or 1.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: OfNat.ofNat, HAdd.hAdd, HMul.hMul, HMod.hMod, HDiv.hDiv, HPow.hPow
13 instances, coercions and auxiliary constants left out.
```
def caRule167 (c_L c_C c_R : ℕ) : ℕ :=
…
```
def OeisA267581.caState The state of the Rule 167 elementary cellular automaton at time $t$ and position $x$ line 50 source
Source documentation
The state of the Rule 167 elementary cellular automaton at time $t$ and position $x$. The initial condition is a single ON cell at $x=0$. $C(t, x)$ is structurally recursive on $t$.

standard axioms: propext what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def caState (t : ℕ) (x : ℤ) : ℕ :=
…
```
def OeisA267581.middleColumnBit The sequence of bits forming the middle column of the CA pattern, $C_{t, 0}$ line 58 source
Source documentation
The sequence of bits forming the middle column of the CA pattern, $C_{t, 0}$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: OeisA267581.caState, OfNat.ofNat, Int
1 instances, coercions and auxiliary constants left out.
```
def middleColumnBit (t : ℕ) : ℕ := caState t 0
```
Mentions: OeisA267581.caState
def OeisA267581.a Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell line 67 source
Source documentation
Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell. The term $a(n)$ is the decimal value of the binary number $C_{0, 0} C_{1, 0} \dots C_{n, 0}$, where $C_{i, 0}$ is the state of the center cell at time $i$. $$a(n) = \sum_{k=0}^n C_{k, 0} \cdot 2^{n-k}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, OeisA267581.middleColumnBit, HPow.hPow, HSub.hSub
12 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA267581.oeisFloorTerm The floor term in the conjectured recurrence relation for A267581 line 75 source
Source documentation
The floor term in the conjectured recurrence relation for A267581. This term, $\lfloor (1/2)^{(2^{n+1} \bmod n)} \rfloor$, simplifies to 1 if $(2^{n+1} \bmod n) = 0$ (i.e., $n \mid 2^{n+1}$), and 0 otherwise. Since the recurrence is only stated for $n \ge 2$, the $n=0$ case is irrelevant to the conjecture.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HMod.hMod, HPow.hPow, HAdd.hAdd
10 instances, coercions and auxiliary constants left out.
```
def oeisFloorTerm (n : ℕ) : ℕ :=
…
```
lemma OeisA267581.a_0 line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA267581.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 OeisA267581.a_1 line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA267581.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 OeisA267581.a_2 line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA267581.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 OeisA267581.a_3 line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA267581.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 OeisA267581.a_4 line 93 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA267581.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 OeisA267581.a_recurrence sorry line 104 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, OeisA267581.a, HSub.hSub, HAdd.hAdd, HMul.hMul, OeisA267581.oeisFloorTerm
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

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