Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Congruence.Hom
import Mathlib.GroupTheory.FinitelyPresentedGroup
import Mathlib.Basic.Complex.Basic

namespace OAI
/-! Section 2 bridge. All algebras here are associative, unital, and may be
noncommutative. The defining congruence is generated by finitely many equations
in a free associative algebra; no commutative finite-presentation API is used. -/
noncomputable section
namespace LittleFinitistic.GroupAlgebraPresentation

variable (R : Type*) [CommRing R] (α : Type*)

/-- A representative word, chosen only to turn a group relator into a polynomial. -/
def word (g : FreeGroup α) : List (α × Bool) :=
  Classical.choose (Quot.exists_rep g)

theorem word_spec (g : FreeGroup α) : FreeGroup.mk (word α g) = g :=
  Classical.choose_spec (Quot.exists_rep g)

def polynomial (g : FreeGroup α) : FreeAlgebra R (α × Bool) :=
  ((word α g).map (FreeAlgebra.ι R)).prod

variable {α R}

theorem eval_polynomial {A : Type*} [Ring A] [Algebra R A]
    (f : α → Aˣ) (F : FreeAlgebra R (α × Bool) →ₐ[R] A)
    (hf : ∀ a b, F (FreeAlgebra.ι R (a,b)) =
      ↑(if b then f a else (f a)⁻¹)) (g : FreeGroup α) :
    F (polynomial R α g) = ↑(FreeGroup.lift f g) := by
  conv_rhs => rw [← word_spec α g]
  simp only [polynomial, map_list_prod, List.map_map, FreeGroup.lift_mk]
  induction word α g with
  | nil => simp
  | cons a l ih =>
    rcases a with ⟨a,b⟩
    simp only [List.map_cons, List.prod_cons, Function.comp_apply, Units.val_mul]
    rw [hf, ih]
    cases b <;> rfl

variable (R α) (rels : Set (FreeGroup α))

abbrev Letter := α × Bool
abbrev EquationIndex := (α × Bool) ⊕ rels

/-- Two inverse equations per generator, and one polynomial per group relator. -/
def equation : EquationIndex α rels → FreeAlgebra R (Letter α)
  | .inl (a,b) => FreeAlgebra.ι R (a,b) * FreeAlgebra.ι R (a,!b)
  | .inr r => polynomial R α r

def relations : RingCon (FreeAlgebra R (Letter α)) :=
  ringConGen (fun x y => ∃ i : EquationIndex α rels, x = equation R α rels i ∧ y = 1)

abbrev PresentedAlgebra := (relations R α rels).Quotient

def quotientMap : FreeAlgebra R (Letter α) →ₐ[R] PresentedAlgebra R α rels :=
  (relations R α rels).mkₐ R

theorem equation_holds (i : EquationIndex α rels) :
    quotientMap R α rels (equation R α rels i) = 1 := by
  exact (relations R α rels).eq.mpr (RingConGen.Rel.of _ _ ⟨i,rfl,rfl⟩)

def generatorUnit (a : α) : (PresentedAlgebra R α rels)ˣ where
  val := quotientMap R α rels (FreeAlgebra.ι R (a,true))
  inv := quotientMap R α rels (FreeAlgebra.ι R (a,false))
  val_inv := by simpa [equation, map_mul] using equation_holds R α rels (.inl (a,true))
  inv_val := by simpa [equation, map_mul] using equation_holds R α rels (.inl (a,false))

theorem quotient_letter (a : α) (b : Bool) :
    quotientMap R α rels (FreeAlgebra.ι R (a,b)) =
      ↑(if b then generatorUnit R α rels a else (generatorUnit R α rels a)⁻¹) := by
  cases b <;> rfl

theorem unit_relator (r : FreeGroup α) (hr : r ∈ rels) :
    FreeGroup.lift (generatorUnit R α rels) r = 1 := by
  apply Units.ext
  rw [← eval_polynomial _ _ (quotient_letter R α rels)]
  exact equation_holds R α rels (.inr ⟨r,hr⟩)

def groupToUnits : PresentedGroup rels →* (PresentedAlgebra R α rels)ˣ :=
  PresentedGroup.toGroup (unit_relator R α rels)

def fromGroupAlgebra : MonoidAlgebra R (PresentedGroup rels) →ₐ[R] PresentedAlgebra R α rels :=
  MonoidAlgebra.lift R _ _ ((Units.coeHom _).comp (groupToUnits R α rels))

def groupAlgebraUnit (a : α) : (MonoidAlgebra R (PresentedGroup rels))ˣ :=
  (MonoidAlgebra.of R (PresentedGroup rels)).toHomUnits (PresentedGroup.of a)

def evaluation : FreeAlgebra R (Letter α) →ₐ[R] MonoidAlgebra R (PresentedGroup rels) :=
  FreeAlgebra.lift R (fun (a,b) =>
    ↑(if b then groupAlgebraUnit R α rels a else (groupAlgebraUnit R α rels a)⁻¹))

theorem evaluation_polynomial (r : FreeGroup α) :
    evaluation R α rels (polynomial R α r) =
      MonoidAlgebra.of R _ (PresentedGroup.mk rels r) := by
  rw [eval_polynomial (groupAlgebraUnit R α rels) _ (by intros; simp [evaluation])]
  have h : FreeGroup.lift (groupAlgebraUnit R α rels) =
      (MonoidAlgebra.of R (PresentedGroup rels)).toHomUnits.comp (PresentedGroup.mk rels) := by
    apply FreeGroup.ext_hom
    intro a
    simp [groupAlgebraUnit, PresentedGroup.of]
  rw [h]
  rfl

theorem evaluation_equation (i : EquationIndex α rels) :
    evaluation R α rels (equation R α rels i) = 1 := by
  rcases i with ⟨a,b⟩ | ⟨r,hr⟩
  · cases b <;> simp [equation, evaluation]
  · rw [equation, evaluation_polynomial]
    have h : PresentedGroup.mk rels r = 1 := by
      exact PresentedGroup.one_of_mem hr
    rw [h, map_one]

def toGroupAlgebra : PresentedAlgebra R α rels →ₐ[R] MonoidAlgebra R (PresentedGroup rels) :=
  (relations R α rels).liftₐ (evaluation R α rels)
    (RingCon.ringConGen_le.mpr (by
      rintro x y ⟨i,rfl,rfl⟩
      exact (evaluation_equation R α rels i).trans (map_one _).symm))

@[simp] theorem toGroupAlgebra_quotient (x : FreeAlgebra R (Letter α)) :
    toGroupAlgebra R α rels (quotientMap R α rels x) = evaluation R α rels x := rfl

@[simp] theorem fromGroupAlgebra_of (a : α) :
    fromGroupAlgebra R α rels (MonoidAlgebra.of R _ (PresentedGroup.of a)) =
      quotientMap R α rels (FreeAlgebra.ι R (a,true)) := by
  simp [fromGroupAlgebra, groupToUnits, generatorUnit]

@[simp] theorem fromGroupAlgebra_inv_of (a : α) :
    fromGroupAlgebra R α rels (MonoidAlgebra.of R _ (PresentedGroup.of a)⁻¹) =
      quotientMap R α rels (FreeAlgebra.ι R (a,false)) := by
  simp [fromGroupAlgebra, groupToUnits, generatorUnit]

@[simp] theorem from_to_letter (a : α) (b : Bool) :
    fromGroupAlgebra R α rels (evaluation R α rels (FreeAlgebra.ι R (a,b))) =
      quotientMap R α rels (FreeAlgebra.ι R (a,b)) := by
  cases b
  · simpa only [evaluation, FreeAlgebra.lift_ι_apply, Bool.false_eq_true, ite_false, groupAlgebraUnit, ← map_inv, MonoidHom.coe_toHomUnits]
      using fromGroupAlgebra_inv_of R α rels a
  · simpa only [evaluation, FreeAlgebra.lift_ι_apply, ite_true, groupAlgebraUnit, MonoidHom.coe_toHomUnits]
      using fromGroupAlgebra_of R α rels a

theorem from_to : (fromGroupAlgebra R α rels).comp (toGroupAlgebra R α rels) =
    AlgHom.id R (PresentedAlgebra R α rels) := by
  have h : ((fromGroupAlgebra R α rels).comp (toGroupAlgebra R α rels)).comp
      (quotientMap R α rels) = quotientMap R α rels := by
    apply FreeAlgebra.hom_ext
    funext ⟨a,b⟩
    exact from_to_letter R α rels a b
  apply AlgHom.ext
  intro x
  obtain ⟨p,rfl⟩ := (relations R α rels).mkₐ_surjective (α := R) x
  exact DFunLike.congr_fun h p

theorem to_from : (toGroupAlgebra R α rels).comp (fromGroupAlgebra R α rels) =
    AlgHom.id R (MonoidAlgebra R (PresentedGroup rels)) := by
  apply (MonoidAlgebra.lift R _ _).symm.injective
  have hext (f g : PresentedGroup rels →* MonoidAlgebra R (PresentedGroup rels))
      (h : ∀ a, f (PresentedGroup.of a) = g (PresentedGroup.of a)) : f = g := by
    have hh : f.comp (PresentedGroup.mk rels) = g.comp (PresentedGroup.mk rels) :=
      FreeGroup.ext_hom _ _ h
    apply MonoidHom.ext
    intro x
    obtain ⟨r,rfl⟩ := PresentedGroup.mk_surjective rels x
    exact DFunLike.congr_fun hh r
  apply hext
  intro a
  change toGroupAlgebra R α rels
    (fromGroupAlgebra R α rels (MonoidAlgebra.of R _ (PresentedGroup.of a))) = _
  rw [fromGroupAlgebra_of, toGroupAlgebra_quotient]
  simp [evaluation, groupAlgebraUnit]

/-- Explicit free associative algebra quotient presenting the group algebra. -/
def quotientEquiv : PresentedAlgebra R α rels ≃ₐ[R] MonoidAlgebra R (PresentedGroup rels) :=
  AlgEquiv.ofAlgHom (toGroupAlgebra R α rels) (fromGroupAlgebra R α rels)
    (to_from R α rels) (from_to R α rels)


/-- A finite presentation in the category of associative unital R-algebras.
The finite generating set is not required to commute. A ring congruence is
closed under multiplication on either side; these are two-sided equations. -/
def HasFinitePresentation (A : Type*) [Ring A] [Algebra R A] : Prop :=
  ∃ (ι κ : Type) (_ : Finite ι) (_ : Finite κ)
    (lhs rhs : κ → FreeAlgebra R ι),
    Nonempty ((ringConGen (fun x y => ∃ i, x = lhs i ∧ y = rhs i)).Quotient ≃ₐ[R] A)

variable {R} in
/-- Transport of a finite noncommutative presentation along an algebra equivalence. -/
theorem HasFinitePresentation.of_equiv {A B : Type*} [Ring A] [Ring B]
    [Algebra R A] [Algebra R B] (h : HasFinitePresentation R A) (e : A ≃ₐ[R] B) :
    HasFinitePresentation R B := by
  obtain ⟨ι,κ,hι,hκ,lhs,rhs,⟨q⟩⟩ := h
  exact ⟨ι,κ,hι,hκ,lhs,rhs,⟨q.trans e⟩⟩

/-- Finitely many group generators and relators give finitely many algebra
letters and equations, and the explicit quotient is the group algebra. -/
theorem presentedGroup_hasFinitePresentation (ι : Type) [Finite ι]
    (s : Set (FreeGroup ι)) [Finite s] :
    HasFinitePresentation R (MonoidAlgebra R (PresentedGroup s)) := by
  exact ⟨Letter ι, EquationIndex ι s, inferInstance, inferInstance,
    equation R ι s, fun _ => 1, ⟨quotientEquiv R ι s⟩⟩

/-- General Section 2 bridge, with no commutativity assumption on the group
or its group algebra and no finite-presentation assumption on the algebra. -/
theorem groupAlgebra_hasFinitePresentation (G : Type*) [Group G]
    [Group.IsFinitelyPresented G] : HasFinitePresentation R (MonoidAlgebra R G) := by
  obtain ⟨n,s,hs,⟨e⟩⟩ := Group.IsFinitelyPresented.exists_mulEquiv_presentedGroup (G := G)
  let : Finite s := hs
  exact (presentedGroup_hasFinitePresentation R (Fin n) s).of_equiv
    (MonoidAlgebra.domCongr R R e.symm)


/-- Polynomial zero relations for the same explicit presentation. -/
def zeroRelations : RingCon (FreeAlgebra R (Letter α)) :=
  ringConGen (fun x y => ∃ i : EquationIndex α rels,
    x = equation R α rels i - 1 ∧ y = 0)

/-- Writing each equation p = 1 as p - 1 = 0 does not change the quotient.
This matches the manuscript notation k⟨x₁,...,x_d⟩/(p₁,...,p_t). -/
theorem relations_eq_zeroRelations : relations R α rels = zeroRelations R α rels := by
  apply le_antisymm
  · apply RingCon.ringConGen_le.mpr
    rintro x y ⟨i,rfl,rfl⟩
    have h : zeroRelations R α rels (equation R α rels i - 1) 0 :=
      RingConGen.Rel.of _ _ ⟨i,rfl,rfl⟩
    have hh := (zeroRelations R α rels).add h ((zeroRelations R α rels).refl 1)
    simpa only [sub_add_cancel, zero_add] using hh
  · apply RingCon.ringConGen_le.mpr
    rintro x y ⟨i,rfl,rfl⟩
    have h : relations R α rels (equation R α rels i) 1 :=
      RingConGen.Rel.of _ _ ⟨i,rfl,rfl⟩
    have hh := (relations R α rels).add h ((relations R α rels).refl (-1))
    simpa only [← sub_eq_add_neg, sub_self] using hh

/-- The explicit presentation is also a quotient by finitely many polynomial
zero relations, in the exact form used by Section 3 of the manuscript. -/
def zeroQuotientEquiv : (zeroRelations R α rels).Quotient ≃ₐ[R]
    MonoidAlgebra R (PresentedGroup rels) :=
  relations_eq_zeroRelations R α rels ▸ quotientEquiv R α rels

end LittleFinitistic.GroupAlgebraPresentation

end

end OAI

	import Mathlib.Algebra.FreeAlgebra
	import Mathlib.RingTheory.Congruence.Hom
	import Mathlib.GroupTheory.FinitelyPresentedGroup
	import Mathlib.Basic.Complex.Basic

	namespace OAI
	/-! Section 2 bridge. All algebras here are associative, unital, and may be
	noncommutative. The defining congruence is generated by finitely many equations
	in a free associative algebra; no commutative finite-presentation API is used. -/
	noncomputable section
	namespace LittleFinitistic.GroupAlgebraPresentation

	variable (R : Type) [CommRing R] (α : Type)

	/-- A representative word, chosen only to turn a group relator into a polynomial. -/
	def word (g : FreeGroup α) : List (α × Bool) :=
	Classical.choose (Quot.exists_rep g)

	theorem word_spec (g : FreeGroup α) : FreeGroup.mk (word α g) = g :=
	Classical.choose_spec (Quot.exists_rep g)

	def polynomial (g : FreeGroup α) : FreeAlgebra R (α × Bool) :=
	((word α g).map (FreeAlgebra.ι R)).prod

	variable {α R}

	theorem eval_polynomial {A : Type*} [Ring A] [Algebra R A]
	(f : α → Aˣ) (F : FreeAlgebra R (α × Bool) →ₐ[R] A)
	(hf : ∀ a b, F (FreeAlgebra.ι R (a,b)) =
	↑(if b then f a else (f a)⁻¹)) (g : FreeGroup α) :
	F (polynomial R α g) = ↑(FreeGroup.lift f g) := by
	conv_rhs => rw [← word_spec α g]
	simp only [polynomial, map_list_prod, List.map_map, FreeGroup.lift_mk]
	induction word α g with
	\| nil => simp
	\| cons a l ih =>
	rcases a with ⟨a,b⟩
	simp only [List.map_cons, List.prod_cons, Function.comp_apply, Units.val_mul]
	rw [hf, ih]
	cases b <;> rfl

	variable (R α) (rels : Set (FreeGroup α))

	abbrev Letter := α × Bool
	abbrev EquationIndex := (α × Bool) ⊕ rels

	/-- Two inverse equations per generator, and one polynomial per group relator. -/
	def equation : EquationIndex α rels → FreeAlgebra R (Letter α)
	\| .inl (a,b) => FreeAlgebra.ι R (a,b) * FreeAlgebra.ι R (a,!b)
	\| .inr r => polynomial R α r

	def relations : RingCon (FreeAlgebra R (Letter α)) :=
	ringConGen (fun x y => ∃ i : EquationIndex α rels, x = equation R α rels i ∧ y = 1)

	abbrev PresentedAlgebra := (relations R α rels).Quotient

	def quotientMap : FreeAlgebra R (Letter α) →ₐ[R] PresentedAlgebra R α rels :=
	(relations R α rels).mkₐ R

	theorem equation_holds (i : EquationIndex α rels) :
	quotientMap R α rels (equation R α rels i) = 1 := by
	exact (relations R α rels).eq.mpr (RingConGen.Rel.of _ _ ⟨i,rfl,rfl⟩)

	def generatorUnit (a : α) : (PresentedAlgebra R α rels)ˣ where
	val := quotientMap R α rels (FreeAlgebra.ι R (a,true))
	inv := quotientMap R α rels (FreeAlgebra.ι R (a,false))
	val_inv := by simpa [equation, map_mul] using equation_holds R α rels (.inl (a,true))
	inv_val := by simpa [equation, map_mul] using equation_holds R α rels (.inl (a,false))

	theorem quotient_letter (a : α) (b : Bool) :
	quotientMap R α rels (FreeAlgebra.ι R (a,b)) =
	↑(if b then generatorUnit R α rels a else (generatorUnit R α rels a)⁻¹) := by
	cases b <;> rfl

	theorem unit_relator (r : FreeGroup α) (hr : r ∈ rels) :
	FreeGroup.lift (generatorUnit R α rels) r = 1 := by
	apply Units.ext
	rw [← eval_polynomial _ _ (quotient_letter R α rels)]
	exact equation_holds R α rels (.inr ⟨r,hr⟩)

	def groupToUnits : PresentedGroup rels →* (PresentedAlgebra R α rels)ˣ :=
	PresentedGroup.toGroup (unit_relator R α rels)

	def fromGroupAlgebra : MonoidAlgebra R (PresentedGroup rels) →ₐ[R] PresentedAlgebra R α rels :=
	MonoidAlgebra.lift R _ _ ((Units.coeHom _).comp (groupToUnits R α rels))

	def groupAlgebraUnit (a : α) : (MonoidAlgebra R (PresentedGroup rels))ˣ :=
	(MonoidAlgebra.of R (PresentedGroup rels)).toHomUnits (PresentedGroup.of a)

	def evaluation : FreeAlgebra R (Letter α) →ₐ[R] MonoidAlgebra R (PresentedGroup rels) :=
	FreeAlgebra.lift R (fun (a,b) =>
	↑(if b then groupAlgebraUnit R α rels a else (groupAlgebraUnit R α rels a)⁻¹))

	theorem evaluation_polynomial (r : FreeGroup α) :
	evaluation R α rels (polynomial R α r) =
	MonoidAlgebra.of R _ (PresentedGroup.mk rels r) := by
	rw [eval_polynomial (groupAlgebraUnit R α rels) _ (by intros; simp [evaluation])]
	have h : FreeGroup.lift (groupAlgebraUnit R α rels) =
	(MonoidAlgebra.of R (PresentedGroup rels)).toHomUnits.comp (PresentedGroup.mk rels) := by
	apply FreeGroup.ext_hom
	intro a
	simp [groupAlgebraUnit, PresentedGroup.of]
	rw [h]
	rfl

	theorem evaluation_equation (i : EquationIndex α rels) :
	evaluation R α rels (equation R α rels i) = 1 := by
	rcases i with ⟨a,b⟩ \| ⟨r,hr⟩
	· cases b <;> simp [equation, evaluation]
	· rw [equation, evaluation_polynomial]
	have h : PresentedGroup.mk rels r = 1 := by
	exact PresentedGroup.one_of_mem hr
	rw [h, map_one]

	def toGroupAlgebra : PresentedAlgebra R α rels →ₐ[R] MonoidAlgebra R (PresentedGroup rels) :=
	(relations R α rels).liftₐ (evaluation R α rels)
	(RingCon.ringConGen_le.mpr (by
	rintro x y ⟨i,rfl,rfl⟩
	exact (evaluation_equation R α rels i).trans (map_one _).symm))

	@[simp] theorem toGroupAlgebra_quotient (x : FreeAlgebra R (Letter α)) :
	toGroupAlgebra R α rels (quotientMap R α rels x) = evaluation R α rels x := rfl

	@[simp] theorem fromGroupAlgebra_of (a : α) :
	fromGroupAlgebra R α rels (MonoidAlgebra.of R _ (PresentedGroup.of a)) =
	quotientMap R α rels (FreeAlgebra.ι R (a,true)) := by
	simp [fromGroupAlgebra, groupToUnits, generatorUnit]

	@[simp] theorem fromGroupAlgebra_inv_of (a : α) :
	fromGroupAlgebra R α rels (MonoidAlgebra.of R _ (PresentedGroup.of a)⁻¹) =
	quotientMap R α rels (FreeAlgebra.ι R (a,false)) := by
	simp [fromGroupAlgebra, groupToUnits, generatorUnit]

	@[simp] theorem from_to_letter (a : α) (b : Bool) :
	fromGroupAlgebra R α rels (evaluation R α rels (FreeAlgebra.ι R (a,b))) =
	quotientMap R α rels (FreeAlgebra.ι R (a,b)) := by
	cases b
	· simpa only [evaluation, FreeAlgebra.lift_ι_apply, Bool.false_eq_true, ite_false, groupAlgebraUnit, ← map_inv, MonoidHom.coe_toHomUnits]
	using fromGroupAlgebra_inv_of R α rels a
	· simpa only [evaluation, FreeAlgebra.lift_ι_apply, ite_true, groupAlgebraUnit, MonoidHom.coe_toHomUnits]
	using fromGroupAlgebra_of R α rels a

	theorem from_to : (fromGroupAlgebra R α rels).comp (toGroupAlgebra R α rels) =
	AlgHom.id R (PresentedAlgebra R α rels) := by
	have h : ((fromGroupAlgebra R α rels).comp (toGroupAlgebra R α rels)).comp
	(quotientMap R α rels) = quotientMap R α rels := by
	apply FreeAlgebra.hom_ext
	funext ⟨a,b⟩
	exact from_to_letter R α rels a b
	apply AlgHom.ext
	intro x
	obtain ⟨p,rfl⟩ := (relations R α rels).mkₐ_surjective (α := R) x
	exact DFunLike.congr_fun h p

	theorem to_from : (toGroupAlgebra R α rels).comp (fromGroupAlgebra R α rels) =
	AlgHom.id R (MonoidAlgebra R (PresentedGroup rels)) := by
	apply (MonoidAlgebra.lift R _ _).symm.injective
	have hext (f g : PresentedGroup rels →* MonoidAlgebra R (PresentedGroup rels))
	(h : ∀ a, f (PresentedGroup.of a) = g (PresentedGroup.of a)) : f = g := by
	have hh : f.comp (PresentedGroup.mk rels) = g.comp (PresentedGroup.mk rels) :=
	FreeGroup.ext_hom _ _ h
	apply MonoidHom.ext
	intro x
	obtain ⟨r,rfl⟩ := PresentedGroup.mk_surjective rels x
	exact DFunLike.congr_fun hh r
	apply hext
	intro a
	change toGroupAlgebra R α rels
	(fromGroupAlgebra R α rels (MonoidAlgebra.of R _ (PresentedGroup.of a))) = _
	rw [fromGroupAlgebra_of, toGroupAlgebra_quotient]
	simp [evaluation, groupAlgebraUnit]

	/-- Explicit free associative algebra quotient presenting the group algebra. -/
	def quotientEquiv : PresentedAlgebra R α rels ≃ₐ[R] MonoidAlgebra R (PresentedGroup rels) :=
	AlgEquiv.ofAlgHom (toGroupAlgebra R α rels) (fromGroupAlgebra R α rels)
	(to_from R α rels) (from_to R α rels)


	/-- A finite presentation in the category of associative unital R-algebras.
	The finite generating set is not required to commute. A ring congruence is
	closed under multiplication on either side; these are two-sided equations. -/
	def HasFinitePresentation (A : Type*) [Ring A] [Algebra R A] : Prop :=
	∃ (ι κ : Type) (_ : Finite ι) (_ : Finite κ)
	(lhs rhs : κ → FreeAlgebra R ι),
	Nonempty ((ringConGen (fun x y => ∃ i, x = lhs i ∧ y = rhs i)).Quotient ≃ₐ[R] A)

	variable {R} in
	/-- Transport of a finite noncommutative presentation along an algebra equivalence. -/
	theorem HasFinitePresentation.of_equiv {A B : Type*} [Ring A] [Ring B]
	[Algebra R A] [Algebra R B] (h : HasFinitePresentation R A) (e : A ≃ₐ[R] B) :
	HasFinitePresentation R B := by
	obtain ⟨ι,κ,hι,hκ,lhs,rhs,⟨q⟩⟩ := h
	exact ⟨ι,κ,hι,hκ,lhs,rhs,⟨q.trans e⟩⟩

	/-- Finitely many group generators and relators give finitely many algebra
	letters and equations, and the explicit quotient is the group algebra. -/
	theorem presentedGroup_hasFinitePresentation (ι : Type) [Finite ι]
	(s : Set (FreeGroup ι)) [Finite s] :
	HasFinitePresentation R (MonoidAlgebra R (PresentedGroup s)) := by
	exact ⟨Letter ι, EquationIndex ι s, inferInstance, inferInstance,
	equation R ι s, fun _ => 1, ⟨quotientEquiv R ι s⟩⟩

	/-- General Section 2 bridge, with no commutativity assumption on the group
	or its group algebra and no finite-presentation assumption on the algebra. -/
	theorem groupAlgebra_hasFinitePresentation (G : Type*) [Group G]
	[Group.IsFinitelyPresented G] : HasFinitePresentation R (MonoidAlgebra R G) := by
	obtain ⟨n,s,hs,⟨e⟩⟩ := Group.IsFinitelyPresented.exists_mulEquiv_presentedGroup (G := G)
	let : Finite s := hs
	exact (presentedGroup_hasFinitePresentation R (Fin n) s).of_equiv
	(MonoidAlgebra.domCongr R R e.symm)


	/-- Polynomial zero relations for the same explicit presentation. -/
	def zeroRelations : RingCon (FreeAlgebra R (Letter α)) :=
	ringConGen (fun x y => ∃ i : EquationIndex α rels,
	x = equation R α rels i - 1 ∧ y = 0)

	/-- Writing each equation p = 1 as p - 1 = 0 does not change the quotient.
	This matches the manuscript notation k⟨x₁,...,x_d⟩/(p₁,...,p_t). -/
	theorem relations_eq_zeroRelations : relations R α rels = zeroRelations R α rels := by
	apply le_antisymm
	· apply RingCon.ringConGen_le.mpr
	rintro x y ⟨i,rfl,rfl⟩
	have h : zeroRelations R α rels (equation R α rels i - 1) 0 :=
	RingConGen.Rel.of _ _ ⟨i,rfl,rfl⟩
	have hh := (zeroRelations R α rels).add h ((zeroRelations R α rels).refl 1)
	simpa only [sub_add_cancel, zero_add] using hh
	· apply RingCon.ringConGen_le.mpr
	rintro x y ⟨i,rfl,rfl⟩
	have h : relations R α rels (equation R α rels i) 1 :=
	RingConGen.Rel.of _ _ ⟨i,rfl,rfl⟩
	have hh := (relations R α rels).add h ((relations R α rels).refl (-1))
	simpa only [← sub_eq_add_neg, sub_self] using hh

	/-- The explicit presentation is also a quotient by finitely many polynomial
	zero relations, in the exact form used by Section 3 of the manuscript. -/
	def zeroQuotientEquiv : (zeroRelations R α rels).Quotient ≃ₐ[R]
	MonoidAlgebra R (PresentedGroup rels) :=
	relations_eq_zeroRelations R α rels ▸ quotientEquiv R α rels

	end LittleFinitistic.GroupAlgebraPresentation

	end

	end OAI