Semigroup with Built-in Associative

Instead of
class Semigroup {
  Element: Type

  mul(x: Element, y: Element): Element

    x: Element,
    y: Element,
    z: Element,
  ): Equal(
    mul(x, mul(y, z)),
    mul(mul(x, y), z)
How about
In which propositional equality in mul_is_associative is transfered the definitional equality of List.
TODO how about Monoid and Group?
Is there a relation between
  • the subclass relation between structures,
  • and an unknown relation between inductive datatypes?