Defining integers inductively in Coq (inductive definitions subject to relations)

Question

In Coq, one can define the natural numbers inductively as follows:

Inductive nat :=
| zero : nat
| succ : nat -> nat.

I would like to know if it's possible to define the integers inductively, in a similar fashion? I can do something like

Inductive int :=
| zero : int
| succ : int -> int
| pred : int -> int.

but then I want to assert in the definition of int that succ (pred x) = x and pred (succ x) = x, and I'm not sure how to do this.

Answer

What you are literally asking for is called a Quotient Inductive Type (QIT) and is not currently supported in Coq even though there is a way to hack around this limitation using private inductive types (see these slides for instance). It goes without saying that it is far from a recommended option (at least in Coq, the situation is different in cubical-agda that do support HITs, a general version of QIT).

In general, Coq does not provide arbitrary quotients. The standard solutions are either to go to setoids (that is types equipped with an equivalence relation and showing that all your constructions preserve these equivalence relation, which is rather heavy) or use specific aspects of the quotient that you want to do (see this paper). The case of integers is actually one of the leading example of that paper.