BlockComplexity.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Block complexity and entropy of a sequence

The block complexity $p(n, a)$ of a sequence $a$ is the number of distinct blocks of $n$ consecutive terms of $a$. The entropy of $a$ is the exponential growth rate of $p(n, a)$.

References:

[AS03] Allouche, Jean-Paul, and Jeffrey Shallit. "Automatic sequences: theory, applications, generalizations." Cambridge University Press, 2003. Chapter 10.
[Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.

Main definitions

SymbolicDynamics.blockComplexity: the number of distinct blocks of length n of a sequence.
SymbolicDynamics.blockEntropy: the entropy of a sequence.

Imports (5)

Imported by 2 files

def SymbolicDynamics.blockComplexity The block complexity $p(n, a)$ of a sequence $a$, that is, the number of distinct blocks $a_k a_{k+1} \cdots a_{k+n-1}$ of $n$ consecutive terms of $a$ line 55 source
Source documentation
The block complexity $p(n, a)$ of a sequence $a$, that is, the number of distinct blocks $a_k a_{k+1} \cdots a_{k+n-1}$ of $n$ consecutive terms of $a$. It is ⊤ when $n \ge 1$ and $a$ takes infinitely many values.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENNReal
Proof uses: Set.encard, Fin, Set.ofPred, Exists, Eq, HAdd.hAdd
4 instances, coercions and auxiliary constants left out.
```
noncomputable def blockComplexity {α : Type*} (a : ℕ → α) (n : ℕ) : ℝ≥0∞ :=
…
```
def SymbolicDynamics.blockEntropy The entropy of a sequence line 64 source
Source documentation
The entropy of a sequence $a$, $$E(a) = \lim_{n \to \infty} \frac{\log p(n, a)}{n}.$$ The limit exists because $n \mapsto \log p(n, a)$ is subadditive, so it agrees with the limsup used here. The value is ⊤ exactly when $a$ takes infinitely many values.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, EReal
Proof uses: Filter.limsup, HDiv.hDiv, ENNReal.log, SymbolicDynamics.blockComplexity, Filter.atTop
12 instances, coercions and auxiliary constants left out.
```
noncomputable def blockEntropy {α : Type*} (a : ℕ → α) : EReal :=
…
```
theorem SymbolicDynamics.blockComplexity_const A constant sequence has exactly one block of each length line 68 source
Source documentation
A constant sequence has exactly one block of each length.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, ENNReal, SymbolicDynamics.blockComplexity, OfNat.ofNat
Proof uses: Set, Fin, Set.ofPred, Exists, HAdd.hAdd, Singleton.singleton, Set.ext, of_eq_true, Iff, Membership.mem, Eq.trans, True, congr, congrArg, congrFun', funext, iff_self, Set.encard, ENat, Set.encard_singleton and 2 more
17 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem blockComplexity_const {α : Type*} (c : α) (n : ℕ) :
    blockComplexity (fun _ ↦ c) n = 1 := by
…
```
Mentions: c, SymbolicDynamics.blockComplexity
theorem SymbolicDynamics.blockEntropy_const A constant sequence has zero entropy line 77 source
Source documentation
A constant sequence has zero entropy.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, EReal, SymbolicDynamics.blockEntropy, Nat, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Filter.limsup, HDiv.hDiv, ENNReal.log, SymbolicDynamics.blockComplexity, Filter.atTop, Filter, funext, ENNReal, SymbolicDynamics.blockComplexity_const, ENNReal.log_one, EReal.zero_div, Filter.limsup_const, eq_self
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem blockEntropy_const {α : Type*} (c : α) : blockEntropy (fun _ ↦ c) = 0 := by
…
```
Mentions: c, SymbolicDynamics.blockEntropy