How to define set in coq without defining set as a list of elements

Question

I am trying to define (1,2,3) as a set of elements in coq. I can define it using list as (1 :: (2 :: (3 :: nil))). Is there any way to define set in coq without using list.

Answer 1

The standard library of coq provides the following finite set modules:

1. Coq.MSets abstracts away the implementation details of the set. For instance, there is an implementation that uses AVL trees and another based on lists.
2. Coq.FSets abstracts away the implementation details of the set; it is a previous version of MSets.
3. Coq.Lists.ListSet is an encoding of lists as sets, which I am including for the sake of completeness

Here is an example on how to define a set with FSets:

Require Import Coq.Structures.OrderedTypeEx.
Require Import Coq.FSets.FSetAVL.

Module NSet := FSetAVL.Make Nat_as_OT.
(* Creates a set with only element 3 inside: *)
Check (NSet.add 3 NSet.empty).