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(** * Definition of arccosine based on arctangent        *)
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(** 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_atan : forall x, 0 < x -> acos x = atan (sqrt (1 - x²) / x).