Notation atan_right_inv := tan_atan (only parsing).

(* Specifications: *)
Lemma atan_opp : forall x, atan (- x) = - atan x.
Lemma derivable_pt_atan : forall x, derivable_pt atan x.
Lemma atan_increasing : forall x y, x < y -> atan x < atan y.
Lemma atan_0 : atan 0 = 0.
Lemma atan_eq0 : forall x, atan x = 0 -> x = 0.
Lemma atan_1 : atan 1 = PI/4.
Lemma atan_tan : forall x, - (PI / 2) < x < PI / 2 -> atan (tan x) = x.
Lemma atan_inv : forall x, (0 < x)%R -> atan (/ x) = (PI / 2 - atan x)%R.
Lemma derive_pt_atan : forall x, derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²).
Lemma derivable_pt_lim_atan : forall x, derivable_pt_lim atan x (/ (1 + x^2)).