set_option structure.eta_thm true set_option structure.proj_mk_thm true structure has_mul [class] (A : Type) := (mul : A → A → A) structure has_add [class] (A : Type) := (add : A → A → A) structure has_one [class] (A : Type) := (one : A) structure has_zero [class] (A : Type) := (zero : A) structure has_inv [class] (A : Type) := (inv : A → A) structure has_neg [class] (A : Type) := (neg : A → A) structure semigroup [class] (A : Type) extends has_mul A := (mul_assoc : ∀a b c, mul (mul a b) c = mul a (mul b c)) structure comm_semigroup [class] (A : Type) extends semigroup A := (mul_comm : ∀a b, mul a b = mul b a) structure left_cancel_semigroup [class] (A : Type) extends semigroup A := (mul_left_cancel : ∀a b c, mul a b = mul a c → b = c) structure right_cancel_semigroup [class] (A : Type) extends semigroup A := (mul_right_cancel : ∀a b c, mul a b = mul c b → a = c) structure add_semigroup [class] (A : Type) extends has_add A := (add_assoc : ∀a b c, add (add a b) c = add a (add b c)) structure add_comm_semigroup [class] (A : Type) extends add_semigroup A := (add_comm : ∀a b, add a b = add b a) structure add_left_cancel_semigroup [class] (A : Type) extends add_semigroup A := (add_left_cancel : ∀a b c, add a b = add a c → b = c) structure add_right_cancel_semigroup [class] (A : Type) extends add_semigroup A := (add_right_cancel : ∀a b c, add a b = add c b → a = c) structure monoid [class] (A : Type) extends semigroup A, has_one A := (one_mul : ∀a, mul one a = a) (mul_one : ∀a, mul a one = a) structure comm_monoid [class] (A : Type) extends monoid A, comm_semigroup A structure add_monoid [class] (A : Type) extends add_semigroup A, has_zero A := (zero_add : ∀a, add zero a = a) (add_zero : ∀a, add a zero = a) structure add_comm_monoid [class] (A : Type) extends add_monoid A, add_comm_semigroup A structure group [class] (A : Type) extends monoid A, has_inv A := (mul_left_inv : ∀a, mul (inv a) a = one) structure comm_group [class] (A : Type) extends group A, comm_monoid A structure add_group [class] (A : Type) extends add_monoid A, has_neg A := (add_left_inv : ∀a, add (neg a) a = zero) structure add_comm_group [class] (A : Type) extends add_group A, add_comm_monoid A structure distrib [class] (A : Type) extends has_mul A, has_add A := (left_distrib : ∀a b c, mul a (add b c) = add (mul a b) (mul a c)) (right_distrib : ∀a b c, mul (add a b) c = add (mul a c) (mul b c)) structure mul_zero_class [class] (A : Type) extends has_mul A, has_zero A := (zero_mul : ∀a, mul zero a = zero) (mul_zero : ∀a, mul a zero = zero) structure zero_ne_one_class [class] (A : Type) extends has_zero A, has_one A := (zero_ne_one : zero ≠ one) structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A, distrib A, mul_zero_class A, zero_ne_one_class A set_option pp.implicit true check @semiring.mul.mk check @semiring.eta