AndrewsCurtis.lean`

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6 declarations, 1 contain sorry, 0 declared as axioms

The Andrews-Curtis conjecture

The conjecture says that every normally generating n-tuple in the free group on n generators is Andrews-Curtis equivalent to the standard free basis.

References:

Imports (1)

FormalConjecturesUtil

abbrev AndrewsCurtis.RelatorTuple An n-tuple of relators in the free group on n generators line 38 source
Source documentation
An n-tuple of relators in the free group on n generators.

no axioms what this means
Statement uses: Nat
Proof uses: Fin, FreeGroup
```
abbrev RelatorTuple (n : ℕ) := Fin n → FreeGroup (Fin n)
```
def AndrewsCurtis.standardRelators The standard relator tuple, consisting of the free generators themselves line 41 source
Source documentation
The standard relator tuple, consisting of the free generators themselves.

no axioms what this means
Statement uses: Nat, AndrewsCurtis.RelatorTuple
Proof uses: Fin, FreeGroup.of
```
def standardRelators (n : ℕ) : RelatorTuple n :=
…
```
Mentions: AndrewsCurtis.RelatorTuple

inductive AndrewsCurtis.Move The elementary Andrews-Curtis moves: inversion or multiplication of relators, interchange of two relators, and conjugation of one relator by an arbitrary word line 48 source

Source documentation

The elementary Andrews-Curtis moves: inversion or multiplication of relators, interchange of two relators, and conjugation of one relator by an arbitrary word.

standard axioms: Classical.choice, Quot.sound, propext what this means

Statement uses: Nat, AndrewsCurtis.RelatorTuple

inductive Move {n : ℕ} : RelatorTuple n → RelatorTuple n → Prop
  | invert (r : RelatorTuple n) (i : Fin n) :
      Move r (Function.update r i (r i)⁻¹)
  | multiply (r : RelatorTuple n) (i j : Fin n) (hij : i ≠ j) :
      Move r (Function.update r i (r i * r j))
  | swap (r : RelatorTuple n) (i j : Fin n) :
      Move r (fun k ↦ r (Equiv.swap i j k))
  | conjugate (r : RelatorTuple n) (i : Fin n) (w : FreeGroup (Fin n)) :
      Move r (Function.update r i (w * r i * w⁻¹))

/-- Andrews-Curtis equivalence is the equivalence relation generated by elementary moves. -/

Mentions: AndrewsCurtis.RelatorTuple

def AndrewsCurtis.Equivalent Andrews-Curtis equivalence is the equivalence relation generated by elementary moves line 59 source
Source documentation
Andrews-Curtis equivalence is the equivalence relation generated by elementary moves.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, AndrewsCurtis.RelatorTuple
Proof uses: Relation.EqvGen, AndrewsCurtis.Move
```
def Equivalent {n : ℕ} : RelatorTuple n → RelatorTuple n → Prop :=
…
```
Mentions: AndrewsCurtis.RelatorTuple
def AndrewsCurtis.NormallyGenerates A relator tuple normally generates the free group when the normal closure of its range is the whole free group line 67 source
Source documentation
A relator tuple normally generates the free group when the normal closure of its range is the whole free group. Equivalently, the associated balanced presentation presents the trivial group.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, AndrewsCurtis.RelatorTuple
Proof uses: Eq, Subgroup, FreeGroup, Fin, Subgroup.normalClosure, Set.range, Top.top
2 instances, coercions and auxiliary constants left out.
```
def NormallyGenerates {n : ℕ} (r : RelatorTuple n) : Prop :=
…
```
Mentions: AndrewsCurtis.RelatorTuple
theorem AndrewsCurtis.andrews_curtis_conjecture The Andrews-Curtis conjecture sorry line 77 source
Source documentation
The Andrews-Curtis conjecture.
Every normally generating n-tuple in the free group of rank n is Andrews-Curtis equivalent to the standard tuple of free generators.

depends on sorry what this means
Statement uses: Nat, AndrewsCurtis.RelatorTuple, AndrewsCurtis.NormallyGenerates, AndrewsCurtis.Equivalent, AndrewsCurtis.standardRelators
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem andrews_curtis_conjecture (n : ℕ) (r : RelatorTuple n)
    (hr : NormallyGenerates r) :
    Equivalent r (standardRelators n) := by
…
```
Mentions: AndrewsCurtis.RelatorTuple, AndrewsCurtis.NormallyGenerates, AndrewsCurtis.Equivalent, AndrewsCurtis.standardRelators