Definition pre_atan (y : R) : {x : R | -PI/2 < x < PI/2 /\ tan x = y}.Definition atan x := let (v, _) := pre_atan x in v.

(* Specifications: *)
Lemma atan_bound : forall x, -PI/2 < atan x < PI/2.
Lemma tan_atan : forall x, tan (atan x) = x.