AutomaticSequence.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

Automatic sequences

A sequence a : ℕ → α is k-automatic if it is the output of a deterministic finite automaton fed the base-k digits of the index. By the kernel criterion of [Eil74] this holds exactly when the k-kernel of a is finite, and that is the definition used here: it needs no automaton, and finiteness of the kernel is what proofs about automatic sequences consume.

For k ≥ 2, IsAutomatic k a already forces the range of a to be finite, by AutomaticSequence.IsAutomatic.finite_range. For k ≤ 1 the k-kernel of a is {a}, so every sequence is k-automatic.

References:

[AS03] Allouche, Jean-Paul, and Jeffrey Shallit. "Automatic Sequences: Theory, Applications, Generalizations." Cambridge University Press, 2003.
[Eil74] Eilenberg, Samuel. "Automata, Languages, and Machines. Vol. A." Academic Press, 1974. Chapter V, Proposition 3.3.

Main definitions

AutomaticSequence.kKernel: the k-kernel of a sequence.
AutomaticSequence.IsAutomatic: k-automaticity, as finiteness of the k-kernel.

Imports (1)

Mathlib.Data.Set.Finite.Basic

Imported by 2 files

def AutomaticSequence.kKernel The k-kernel of a sequence a is the set of subsequences $(a_{k^e n + r})_{n \ge 0}$ with $e \ge 0$ and $0 \le r < k^e$ line 54 source
Source documentation
The k-kernel of a sequence a is the set of subsequences $(a_{k^e n + r})_{n \ge 0}$ with $e \ge 0$ and $0 \le r < k^e$.

no axioms what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, And, LT.lt, HPow.hPow, Eq, HAdd.hAdd, HMul.hMul
8 instances, coercions and auxiliary constants left out.
```
def kKernel (k : ℕ) (a : ℕ → α) : Set (ℕ → α) :=
…
```
theorem AutomaticSequence.self_mem_kKernel A sequence is a member of its own k-kernel, via e = r = 0 line 58 source
Source documentation
A sequence is a member of its own k-kernel, via e = r = 0.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Set, AutomaticSequence.kKernel
Proof uses: Exists.intro, Exists, And, LT.lt, HPow.hPow, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, And.intro, of_eq_true, Eq.trans, True, congrArg, funext, congrFun', Nat.one_mul, eq_self
20 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem self_mem_kKernel (k : ℕ) (a : ℕ → α) : a ∈ kKernel k a :=
…
```
Mentions: AutomaticSequence.kKernel
def AutomaticSequence.IsAutomatic A sequence a is k-automatic if its k-kernel is finite line 66 source
Source documentation
A sequence a is k-automatic if its k-kernel is finite. By [Eil74] this is equivalent to a being generated by a deterministic finite automaton reading the base-k digits of the index.

no axioms what this means
Statement uses: Nat
Proof uses: Set.Finite, AutomaticSequence.kKernel
```
def IsAutomatic (k : ℕ) (a : ℕ → α) : Prop := (kKernel k a).Finite
```
Mentions: AutomaticSequence.kKernel
theorem AutomaticSequence.IsAutomatic.finite_range A k-automatic sequence with k ≥ 2 takes finitely many values: a m is the value at 0 of the kernel element n ↦ a (k ^ m * n + m) line 72 source
Source documentation
A k-automatic sequence with k ≥ 2 takes finitely many values: a m is the value at 0 of the kernel element n ↦ a (k ^ m * n + m).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, AutomaticSequence.IsAutomatic, Set.Finite, Set.range
Proof uses: Set.Finite.subset, Set.image, AutomaticSequence.kKernel, Set.Finite.image, Iff.mpr, Set, Membership.mem, Set.range_subset_iff, Exists.intro, And, Eq, HAdd.hAdd, HMul.hMul, HPow.hPow, And.intro, Exists, LT.lt, Nat.lt_pow_self, rfl, of_eq_true and 6 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem IsAutomatic.finite_range {k : ℕ} {a : ℕ → α} (hk : 2 ≤ k) (h : IsAutomatic k a) :
    (Set.range a).Finite :=
…
```
Mentions: AutomaticSequence.IsAutomatic
theorem AutomaticSequence.kKernel_const The k-kernel of a constant sequence is a singleton line 79 source
Source documentation
The k-kernel of a constant sequence is a singleton.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Eq, Set, AutomaticSequence.kKernel, Singleton.singleton
Proof uses: Set.ext, Eq.mpr, Iff, Membership.mem, Exists, And, LT.lt, HPow.hPow, id, congrArg, Iff.intro, Exists.intro, OfNat.ofNat, And.intro, of_eq_true
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem kKernel_const (k : ℕ) (c : α) : kKernel k (fun _ : ℕ => c) = {fun _ => c} := by
…
```
Mentions: c, AutomaticSequence.kKernel
theorem AutomaticSequence.isAutomatic_const A constant sequence is k-automatic line 85 source
Source documentation
A constant sequence is k-automatic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, AutomaticSequence.IsAutomatic
Proof uses: Eq.mpr, Set.Finite, AutomaticSequence.kKernel, id, Eq, congrArg, AutomaticSequence.IsAutomatic.eq_1, Singleton.singleton, Set, AutomaticSequence.kKernel_const, Set.finite_singleton
1 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem isAutomatic_const (k : ℕ) (c : α) : IsAutomatic k (fun _ => c) := by
…
```
Mentions: c, AutomaticSequence.IsAutomatic