# Polymorphic ClassesΒΆ

Polymorphic classes can be used to systematically introduce notation systems representing algebras.

For example let us start with a sketch of the library class Eq which introduces equivalence relations:

```
class Eq[t] {
virtual fun == : t * t -> bool;
virtual fun != (x:t,y:t):bool => not (x == y);
axiom reflex(x:t): x == x;
axiom sym(x:t, y:t): (x == y) == (y == x);
axiom trans(x:t, y:t, z:t): x == y and y == z implies x == z;
fun eq(x:t, y:t)=> x == y;
fun ne(x:t, y:t)=> x != y;
}
```

The class has a single type parameter t. Two virtual functions are introduced first, == for equivalence, and != for inequivalence.

The type of the == function is given but it is not defined. We only have the interface.

The inequivalence is defined as the negation of equivalence.

Next we have three axioms which specify the required semantics.

The first axiom, reflexivity, says a value is equivalent to itself.

The second axiom, symmetry, says a if x is equivalent to y, then y is equivalent to x.

The third axiom, transitivity says if x is equivalent to y, and y is equivalent to z, then z is equivalent to z too.

In this case these three laws are a complete specification, and, the laws are independent, in that one cannot be deduced from another. These two properties mean that our rules are in fact mathematical axioms.

Felix does not require axioms actually be axioms. However, simple rules which can be derived from the stated axioms can be specified by lemma. The idea is that lemmas can be so eaily proven from the axioms that an automatic theorem prover can do it, without any further assistance.

You can also a theorem which is a rule that can be proven from the axioms, but where the proof requires a human, or a human with a proof assistant, to establish its correctness.

Finally in our class, we have two named function defined in terms of the virtual functions. Notice these functions are not virtual.