diff --git a/library/algebra/group.lean b/library/algebra/group.lean index bf1e19637..3426a3f08 100644 --- a/library/algebra/group.lean +++ b/library/algebra/group.lean @@ -252,22 +252,24 @@ section group iff.intro eq_mul_inv_of_mul_eq mul_eq_of_eq_mul_inv definition group.to_left_cancel_semigroup [instance] [coercion] : left_cancel_semigroup A := - left_cancel_semigroup.mk (@group.mul A s) (@group.mul_assoc A s) - (take a b c, + ⦃ left_cancel_semigroup, + mul_left_cancel := take a b c, assume H : a * b = a * c, calc b = a⁻¹ * (a * b) : inv_mul_cancel_left ... = a⁻¹ * (a * c) : H - ... = c : inv_mul_cancel_left) + ... = c : inv_mul_cancel_left, + using s ⦄ definition group.to_right_cancel_semigroup [instance] [coercion] : right_cancel_semigroup A := - right_cancel_semigroup.mk (@group.mul A s) (@group.mul_assoc A s) - (take a b c, + ⦃ right_cancel_semigroup, + mul_right_cancel := take a b c, assume H : a * b = c * b, calc a = (a * b) * b⁻¹ : mul_inv_cancel_right ... = (c * b) * b⁻¹ : H - ... = c : mul_inv_cancel_right) + ... = c : mul_inv_cancel_right, + using s ⦄ end group @@ -380,23 +382,25 @@ section add_group definition add_group.to_left_cancel_semigroup [instance] [coercion] : add_left_cancel_semigroup A := - add_left_cancel_semigroup.mk add_group.add add_group.add_assoc - (take a b c, + ⦃ add_left_cancel_semigroup, + add_left_cancel := take a b c, assume H : a + b = a + c, calc b = -a + (a + b) : !neg_add_cancel_left⁻¹ ... = -a + (a + c) : H - ... = c : neg_add_cancel_left) + ... = c : neg_add_cancel_left, + using s ⦄ definition add_group.to_add_right_cancel_semigroup [instance] [coercion] : add_right_cancel_semigroup A := - add_right_cancel_semigroup.mk add_group.add add_group.add_assoc - (take a b c, + ⦃ add_right_cancel_semigroup, + add_right_cancel := take a b c, assume H : a + b = c + b, calc a = (a + b) + -b : !add_neg_cancel_right⁻¹ ... = (c + b) + -b : H - ... = c : add_neg_cancel_right) + ... = c : add_neg_cancel_right, + using s ⦄ /- sub -/