DFA.lean`

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

5 declarations, 0 contain sorry, 0 declared as axioms

Synchronizing words for DFAs

A synchronizing word (or reset word) for a DFA M = (Q, Σ, δ) is a word w ∈ Σ* such that reading w from any state always leads to the same single state. A DFA is synchronizing if it admits at least one synchronizing word.

These notions support the Černý conjecture (see FormalConjectures.Wikipedia.CernyConjecture).

Imports (1)

Mathlib.Computability.DFA

Imported by 1 file

FormalConjecturesForMathlib.lean

def DFA.IsSynchronizingWord A word w is a synchronizing word (or reset word) for a DFA M if reading w from any state leads to the same single state line 38 source
Source documentation
A word w is a synchronizing word (or reset word) for a DFA M if reading w from any state leads to the same single state.

no axioms what this means
Statement uses: DFA, List
Proof uses: Exists, Eq, DFA.evalFrom
```
def IsSynchronizingWord (M : DFA α σ) (w : List α) : Prop :=
…
```
def DFA.IsSynchronizing A DFA M is synchronizing if it has at least one synchronizing word line 42 source
Source documentation
A DFA M is synchronizing if it has at least one synchronizing word.

no axioms what this means
Statement uses: DFA
Proof uses: Exists, List, DFA.IsSynchronizingWord
```
def IsSynchronizing (M : DFA α σ) : Prop :=
…
```
theorem DFA.isSynchronizingWord_nil The empty word is a synchronizing word for M exactly when all states of M collapse to a single one (equivalently, σ is a nonempty subsingleton) line 48 source
Source documentation
The empty word is a synchronizing word for M exactly when all states of M collapse to a single one (equivalently, σ is a nonempty subsingleton).

standard axioms: propext what this means
Statement uses: DFA, Iff, DFA.IsSynchronizingWord, List.nil, Exists, Eq
Proof uses: of_eq_true, iff_self

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isSynchronizingWord_nil (M : DFA α σ) :
    M.IsSynchronizingWord [] ↔ ∃ p : σ, ∀ q : σ, q = p := by
…
```
theorem DFA.IsSynchronizingWord.append Documented declaration: append line 55 source
Source documentation
Appending any word v to a synchronizing word w yields a synchronizing word: reading w already drives every state to a single state p, and reading v afterwards sends p to M.evalFrom p v regardless of the starting state.

standard axioms: Quot.sound, propext what this means
Statement uses: DFA, List, DFA.IsSynchronizingWord, HAppend.hAppend
Proof uses: Eq, DFA.evalFrom, Exists.intro, Eq.mpr, id, congrArg, DFA.evalFrom_of_append, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem IsSynchronizingWord.append {M : DFA α σ} {w : List α}
    (hw : M.IsSynchronizingWord w) (v : List α) : M.IsSynchronizingWord (w ++ v) := by
…
```
theorem DFA.isSynchronizing_of_subsingleton A DFA whose state type is a nonempty subsingleton (i.e. a single-state DFA) is trivially synchronizing: the empty word already synchronizes it line 63 source
Source documentation
A DFA whose state type is a nonempty subsingleton (i.e. a single-state DFA) is trivially synchronizing: the empty word already synchronizes it.

standard axioms: propext what this means
Statement uses: Nonempty, Subsingleton, DFA, DFA.IsSynchronizing
Proof uses: Exists.intro, List, DFA.IsSynchronizingWord, List.nil, Iff.mpr, Exists, Eq, DFA.isSynchronizingWord_nil, Subsingleton.elim
1 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem isSynchronizing_of_subsingleton [Nonempty σ] [Subsingleton σ] (M : DFA α σ) :
    M.IsSynchronizing := by
…
```