@CommMonoid.mk.{u_1} : {α : Type u_1} →
  [toMonoid : Monoid.{u_1} α] →
    (∀ (a b : α),
        @Eq.{u_1 + 1} α
          (@HMul.hMul.{u_1, u_1, u_1} α α α
            (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α (@Monoid.toSemigroup.{u_1} α toMonoid))) a b)
          (@HMul.hMul.{u_1, u_1, u_1} α α α
            (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α (@Monoid.toSemigroup.{u_1} α toMonoid))) b a)) →
      CommMonoid.{u_1} α
def CommMonoid.toCommSemigroup.{u} : {α : Type u} → [self : CommMonoid.{u} α] → CommSemigroup.{u} α :=
fun (α : Type u) [self : CommMonoid.{u} α] =>
  @CommSemigroup.mk.{u} α (@Monoid.toSemigroup.{u} α (@CommMonoid.toMonoid.{u} α self))
    (@CommMonoid.mul_comm.{u} α self)
@CommGroup.mk.{u_1} : {α : Type u_1} →
  [toGroup : Group.{u_1} α] →
    (∀ (a b : α),
        @Eq.{u_1 + 1} α
          (@HMul.hMul.{u_1, u_1, u_1} α α α
            (@instHMul.{u_1} α
              (@Semigroup.toMul.{u_1} α (@Monoid.toSemigroup.{u_1} α (@Group.toMonoid.{u_1} α toGroup))))
            a b)
          (@HMul.hMul.{u_1, u_1, u_1} α α α
            (@instHMul.{u_1} α
              (@Semigroup.toMul.{u_1} α (@Monoid.toSemigroup.{u_1} α (@Group.toMonoid.{u_1} α toGroup))))
            b a)) →
      CommGroup.{u_1} α
def CommGroup.toCommMonoid.{u} : {α : Type u} → [self : CommGroup.{u} α] → CommMonoid.{u} α :=
fun (α : Type u) [self : CommGroup.{u} α] =>
  @CommMonoid.mk.{u} α (@Group.toMonoid.{u} α (@CommGroup.toGroup.{u} α self)) (@CommGroup.mul_comm.{u} α self)
Field.mk : {α : Type u_1} → [toDivisionRing : DivisionRing α] → (∀ (a b : α), a * b = b * a) → Field α
def Field.toDivisionRing.{u} : {α : Type u} → [self : Field.{u} α] → DivisionRing.{u} α :=
fun (α : Type u) [self : Field.{u} α] => self.1
def Field.toCommRing.{u} : {α : Type u} → [self : Field.{u} α] → CommRing.{u} α :=
fun (α : Type u) [self : Field.{u} α] =>
  @CommRing.mk.{u} α (@DivisionRing.toRing.{u} α (@Field.toDivisionRing.{u} α self)) (@Field.mul_comm.{u} α self)