Inheritance

When defining classes, you can inherit methods from other classes or specialisations thereof. For example here is a total order:

class Tord[t]{
  inherit Eq[t];

  virtual fun < : t * t -> bool;
  fun lt (x:t,y:t): bool=> x < y;

  axiom trans(x:t, y:t, z:t): x < y and y < z implies x < z;
  axiom antisym(x:t, y:t): x < y or y < x or x == y;
  axiom reflex(x:t, y:t): x < y and y <= x implies x == y;
  axiom totality(x:t, y:t): x <= y or y <= x;

  fun >(x:t,y:t):bool => y < x;
  fun gt(x:t,y:t):bool => y < x;

  fun <= (x:t,y:t):bool => not (y < x);
  fun le (x:t,y:t):bool => not (y < x);

  fun >= (x:t,y:t):bool => not (x < y);
  fun ge (x:t,y:t):bool => not (x < y);

  fun max(x:t,y:t):t=> if x < y then y else x endif;
  fun \vee(x:t,y:t) => max (x,y);

  fun min(x:t,y:t):t => if x < y then x else y endif;
  fun \wedge(x:t,y:t):t => min (x,y);
}

The inherit statement pulls in the methods of Eq so you can write:

println$ Tord[int]::eq(1,2);

and expect it to work. However when instantiating a total order you cannot provide a definition for inherited methods, you must provide the instance for the original class:

instance Eq[int] {
  fun == : int * int -> bool = "$1==$2";
}
instance Tord[int] {
  fun < : int * int -> bool = "$1<$2";
}

Although in this case, we inherited Eq[t], for all t, we could have inherited Eq[int], for example. Instances can only be provided for a class, not a specialisation, because the instances are themselves defining specialisations.