Rule30.lean`

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8 declarations, 2 contain sorry, 0 declared as axioms

The Rule 30 Prize Problems

Rule 30 is the elementary cellular automaton with local update $c' = l \oplus (c \lor r)$, the Boolean rule numbered $30$ by Wolfram. Started from a single black cell on a bi-infinite row, its center column t ↦ (state t) 0 looks random — it was long Mathematica's default pseudorandom generator — yet nothing about that randomness is proven. In 2019 Wolfram offered the Rule 30 Prizes for three questions about it; we formalize the first two.

Problem 1. Is the center column non-periodic (never eventually periodic)?
Problem 2. Does each color occur on average equally often, i.e. does the running average of the values converge to $1/2$?

Problem 3 — whether computing the $n$-th cell requires at least $O(n)$ work — is omitted. It is model-relative and has no canonical model-independent phrasing. All three are open.

References:

Announcing the Rule 30 Prizes, Stephen Wolfram, 2019.
Rule 30 Prizes.
Wikipedia: Rule 30.

Imports (1)

FormalConjecturesUtil

def Rule30.step The Rule 30 local update on a bi-infinite row row : ℤ → Bool line 49 source
Source documentation
The Rule 30 local update on a bi-infinite row row : ℤ → Bool. The new value at position $i$ is row (i-1) ⊕ (row i ∨ row (i+1)), i.e. Wolfram's rule $30$.

no axioms what this means
Statement uses: Int, Bool
Proof uses: Bool.xor, HSub.hSub, OfNat.ofNat, Bool.or, HAdd.hAdd
5 instances, coercions and auxiliary constants left out.
```
def step (row : ℤ → Bool) : ℤ → Bool :=
…
```
def Rule30.state The Rule 30 evolution from the single-black-cell initial condition: state 0 is black exactly at position $0$, and each subsequent row is obtained by applying step line 54 source
Source documentation
The Rule 30 evolution from the single-black-cell initial condition: state 0 is black exactly at position $0$, and each subsequent row is obtained by applying step.

no axioms what this means
Statement uses: Nat, Int, Bool
2 instances, coercions and auxiliary constants left out.
```
def state : ℕ → ℤ → Bool
  | 0     => fun i => decide (i = 0)
  | t + 1 => step (state t)

@[simp, category API, AMS 37 68]
```
Mentions: Rule30.step, AMS
theorem Rule30.state_zero line 59 source
no axioms what this means
Statement uses: Int, Eq, Bool, Rule30.state, OfNat.ofNat, Nat, Decidable.decide
Proof uses: rfl
4 instances, coercions and auxiliary constants left out.
theorem Rule30.state_succ line 62 source
no axioms what this means
Statement uses: Nat, Eq, Int, Bool, Rule30.state, HAdd.hAdd, OfNat.ofNat, Rule30.step
Proof uses: rfl
6 instances, coercions and auxiliary constants left out.
def Rule30.centerColumn The center column of Rule 30: the value of the single central cell (position $0$) at time $t$ line 66 source
Source documentation
The center column of Rule 30: the value of the single central cell (position $0$) at time $t$.

no axioms what this means
Statement uses: Nat, Bool
Proof uses: Rule30.state, OfNat.ofNat, Int
1 instances, coercions and auxiliary constants left out.
```
def centerColumn (t : ℕ) : Bool := state t 0
…
```
Mentions: Rule30.state
theorem Rule30.centerColumn_prefix Sanity check: the first center-column values reproduce the known Rule 30 center column (OEIS A051023) line 73 source
Source documentation
Sanity check: the first center-column values reproduce the known Rule 30 center column (OEIS A051023). Unlike state_zero/state_succ, this exercises the composed evolution, guarding against a wrong seed, a swapped xor/||, or an off-by-one neighbour index.

L4DEL explanation
Sanity check: the first center-column values reproduce the known Rule 30 center column (OEIS A051023). Unlike state_zero/state_succ, this exercises the composed evolution, guarding against a wrong seed, a swapped xor/||, or an off-by-one neighbour index.
Sources and attribution: Pinned Lean source (1646ca16afd6)

no axioms what this means
Statement uses: Eq, List, Bool, List.map, Nat, Rule30.centerColumn, List.range, OfNat.ofNat, List.cons, Bool.true, Bool.false, List.nil
Proof uses: of_decide_eq_true, id, Decidable.decide, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem centerColumn_prefix :
    (List.range 8).map centerColumn = [true, true, false, true, true, true, false, false] := by
…
```
Mentions: Rule30.centerColumn
theorem Rule30.centerColumn_not_eventually_periodic Rule 30 Prize, Problem 1 (non-periodicity) sorry line 81 source
Source documentation
Rule 30 Prize, Problem 1 (non-periodicity). The center column of Rule 30 is not eventually periodic: there is no positive period $p$ and threshold $N$ past which the column repeats with period $p$.

depends on sorry what this means
Statement uses: Iff, True, Not, Exists, Nat, And, LT.lt, OfNat.ofNat, LE.le, Eq, Bool, Rule30.centerColumn, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem centerColumn_not_eventually_periodic :
    answer(sorry) ↔
      ¬ ∃ p : ℕ, 0 < p ∧ ∃ N : ℕ, ∀ t : ℕ, N ≤ t → centerColumn (t + p) = centerColumn t := by
…
```
Mentions: Rule30.centerColumn
theorem Rule30.centerColumn_frequency_half Rule 30 Prize, Problem 2 (equal frequency) sorry line 91 source
Source documentation
Rule 30 Prize, Problem 2 (equal frequency). Each color occurs on average equally often in the center column: the set of times at which it is black has natural density $1/2$. This is Wolfram's phrasing that the discrete limit of $\mathrm{Total}[c[t]]/t$ as $t \to \infty$ is $1/2$.

depends on sorry what this means
Statement uses: Iff, True, Set.HasDensity, Nat, Set.ofPred, Eq, Bool, Rule30.centerColumn, Bool.true, HDiv.hDiv, Real, OfNat.ofNat, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem centerColumn_frequency_half :
    answer(sorry) ↔ {t : ℕ | centerColumn t}.HasDensity (1 / 2) := by
…
```
Mentions: Rule30.centerColumn