New Post: Express relations and their properties

Hello:

I want to represent some relations in Dafny and state that they follow certain properties like transitivity, antisymmetry or totalness.

For example, suppose we want to declare a binary relation on a type T which is a total preorder relation. We may write the following predicate along with its three properties (Reflexive is superfluous because Total is harder, unless == does something strange, I think).

predicate Leq<T>(x : T, y : T)
    // Reflexive
    ensures x == y ==> Leq(x, y)

    // Transitive
    ensures forall z : T :: Leq(x, z) && Leq(z, y) ==> Leq(x, y)

    // Total
    ensures Leq(x, y) || Leq(y, x)


method Main()
{
    var a : int, b : int, c : int;

    assume Leq(a, b);
    assume Leq(b, c);

    // Transitivity
    assert Leq(a, c);

    // Reflexivity
    assert Leq(a, a);

    var x : int, y : int;

    assume !Leq(x, y);

    // Totalness
    assert Leq(y, x);
}

This function can be used in any other function or method taking advantage of its properties, as in this example. But the function verification itself fails and Dafny complains about termination in the second and third postconditions.

A lemma might be used instead but it seems to be quite uncomfortable. I want to abstract from the particular relation definition. What is the better way to represent that in Dafny?

Thanks in advance.