English
The free abelian group on α can be endowed with a ring structure extending the monoid structure on α; the ring operations are defined by the lift construction, yielding a ring on FreeAbelianGroup α.
Русский
Свободная абелева группа на α может быть снабжена структурой кольца, расширяющей моноидальную структуру α; операции умножения/сложения задаются через конструирование lift, образуя кольцо на FreeAbelianGroup α.
LaTeX
$$등$$
Lean4
instance ring : Ring (FreeAbelianGroup α) :=
{ FreeAbelianGroup.nonUnitalRing _,
FreeAbelianGroup.one
_ with
mul_one := fun x ↦ by
rw [mul_def, one_def, lift_apply_of]
refine FreeAbelianGroup.induction_on x rfl (fun L ↦ ?_) (fun L ih ↦ ?_) fun x1 x2 ih1 ih2 ↦ ?_
· rw [lift_apply_of, mul_one]
· rw [map_neg, ih]
· rw [map_add, ih1, ih2]
one_mul := fun x ↦ by
simp_rw [mul_def, one_def, lift_apply_of]
refine FreeAbelianGroup.induction_on x rfl ?_ ?_ ?_
· intro L
rw [lift_apply_of, one_mul]
· intro L ih
rw [map_neg, ih]
· intro x1 x2 ih1 ih2
rw [map_add, ih1, ih2] }
