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

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

Definition acos x :=
  if Rle_dec x (-1) then PI else
  if Rle_dec 1 x then 0 else
  PI/2 - atan (x/sqrt(1 - x²)).

Lemma acos_cos : forall x, 0 <= x <= PI -> acos (cos x) = x.