AutomaticNumber.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

3 declarations, 0 contain sorry, 0 declared as axioms

Automatic numbers

A real number is automatic in base b if its base-b digit sequence is generated by a finite automaton. Two readings of this are in use, and they differ:

the classical one, that the digit sequence is b-automatic;
the broad one, that the digit sequence is k-automatic for some k ≥ 2, which need not be b.

The broad reading is the weaker hypothesis on the number, so a theorem that rules out automatic numbers is stronger when stated with it. Adamczewski and Faverjon prove their two-base rigidity results in the broad form.

Only the digits after the radix point are used, as in NormalNumber.digitSeq. The integer part of a real number contributes finitely many digits, so it does not affect automaticity.

References:

AF20 Adamczewski, Boris, and Colin Faverjon. "Mahler's method in several variables and finite automata." arXiv:2012.08283 (2020). Appendix A.
[AS03] Allouche, Jean-Paul, and Jeffrey Shallit. "Automatic Sequences: Theory, Applications, Generalizations." Cambridge University Press, 2003.

Main definitions

AutomaticSequence.IsAutomaticInBase: the classical reading.
AutomaticSequence.IsBroadlyAutomaticInBase: the broad reading.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def AutomaticSequence.IsAutomaticInBase The base-b expansion of ξ is generated by a finite automaton, in the classical sense: its digit sequence is b-automatic line 59 source
Source documentation
The base-b expansion of ξ is generated by a finite automaton, in the classical sense: its digit sequence is b-automatic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: AutomaticSequence.IsAutomatic, NormalNumber.digitSeq
```
def IsAutomaticInBase (b : ℕ) (ξ : ℝ) : Prop := IsAutomatic b (NormalNumber.digitSeq b ξ)
```
Mentions: AutomaticSequence.IsAutomatic, NormalNumber.digitSeq
def AutomaticSequence.IsBroadlyAutomaticInBase The base-b expansion of ξ is generated by a finite automaton, in the broad sense: its digit sequence is k-automatic for some k ≥ 2, not necessarily for k = b line 65 source
Source documentation
The base-b expansion of ξ is generated by a finite automaton, in the broad sense: its digit sequence is k-automatic for some k ≥ 2, not necessarily for k = b.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Exists, And, LE.le, OfNat.ofNat, AutomaticSequence.IsAutomatic, NormalNumber.digitSeq
2 instances, coercions and auxiliary constants left out.
```
def IsBroadlyAutomaticInBase (b : ℕ) (ξ : ℝ) : Prop :=
…
```
theorem AutomaticSequence.IsAutomaticInBase.broad The classical reading implies the broad one, by taking k = b line 69 source
Source documentation
The classical reading implies the broad one, by taking k = b.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, OfNat.ofNat, AutomaticSequence.IsAutomaticInBase, AutomaticSequence.IsBroadlyAutomaticInBase
Proof uses: Exists.intro, And, AutomaticSequence.IsAutomatic, NormalNumber.digitSeq, And.intro
4 instances, coercions and auxiliary constants left out.
```
theorem IsAutomaticInBase.broad {b : ℕ} {ξ : ℝ} (hb : 2 ≤ b) (h : IsAutomaticInBase b ξ) :
    IsBroadlyAutomaticInBase b ξ := ⟨b, hb, h⟩
…
```
Mentions: AutomaticSequence.IsAutomaticInBase, AutomaticSequence.IsBroadlyAutomaticInBase