(*********************************************************)
(** * Definition of arcsine based on arctangent          *)
(*********************************************************)

(** asin is defined by cases so that it is defined in the full range from -1 .. 1 *)

Definition asin x :=
  if Rle_dec x (-1) then - (PI / 2) else
  if Rle_dec 1 x then PI / 2 else
  atan (x / sqrt (1 - x²)).

Lemma derive_pt_asin : forall (x : R) (Hxrange : -1 < x < 1), derive_pt asin x (derivable_pt_asin x Hxrange) = 1 / sqrt (1 - x²).