(*****************************************************)
(**                   Addition                       *)

Definition def_add (x y : N.t) := recursion y (fun _ => S) x.Local Infix "+++" := def_add (at level 50, left associativity).#[global]
Instance def_add_wd : Proper (N.eq ==> N.eq ==> N.eq) def_add.

(* Specifications: *)
Theorem def_add_0_l : forall y, 0 +++ y == y.
Theorem def_add_succ_l : forall x y, S x +++ y == S (x +++ y).
Theorem def_add_add : forall n m, n +++ m == n + m.