Set ADT - algebraic specification

This version uses axioms to prevent duplicate items.

ADT Set
    uses Any, Boolean, Integer
    defines Set<T: Any>
    operations:
        new:  ---> Set<T>
        size: Set<T> ---> Integer
        isEmpty: Set<T> ---> Boolean
        add: Set<T> x T ---> Set<T>
        remove: Set<T> x T ---> Set<T>
        contains: Set<T> x T ---> Boolean
    axioms:
        size(new()) = 0
        isEmpty(b) = if size(b) = 0 then true else false
        remove(add(s,t), t) = s
        contains(add(s,t),v) = if t = v then true else contains(s,v)
        contains(remove(s,t),v) = if t = v then false else contains(s,v)

        if contains(s,t) then add(s,t) = s
        if !contains(s,t) then remove(s,t) = s
        if !contains(s,t) then size(add(s,t)) = size(s) + 1
        if contains(s,t) then size(remove(s,t)) = size(s) - 1



This version uses pre-conditions to prevent duplicates.

ADT Set
    uses Any, Boolean, Integer
    defines Set<T: Any>
    operations:
        new: ---> Set<T>
        size: Set<T> ---> Integer
        add: Set<T> x T -/-> Set<T>
        remove: Set<T> x T -/-> Set<T>
        contains: Set<T> x T ---> Boolean
    preconditions:
        add(s,t): contains(s,t) = false
        remove(s,t): contains(s,t) = true
    axioms:
        size(new()) = 0
        isEmpty(b) = if size(b) = 0 then true else false
        size(add(s,t)) = size(s) + 1
        size(remove(s,t)) = size(s) - 1
        contains(new(),t) = false
        contains(add(s,t),v) = if t = v then true else contains(s,v)
        contains(remove(s,t),v) = if t = v then false else contains(s,v)