(* Proving a few formulas in the style of John Machin to compute Pi *)

Definition atan_sub u v := (u - v)/(1 + u * v).

Lemma atan_sub_correct : forall u v, 1 + u * v <> 0 -> -PI/2 < atan u - atan v < PI/2 -> -PI/2 < atan (atan_sub u v) < PI/2 -> atan u = atan v + atan (atan_sub u v).