import types.sigma types.prod import algebra.binary algebra.group open eq eq.ops namespace path_algebra variable {A : Type} 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 : Πa, mul zero a = zero) (mul_zero : Πa, 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 end path_algebra