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

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

(* Specifications: *)
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).
Lemma tech : forall x y , -1 <= x <= 1 -> -1 < y < 1 -> -PI/2 < atan x - atan y < PI/2.
Lemma Machin_2_3 : PI/4 = atan(/2) + atan(/3).
Lemma Machin_4_5_239 : PI/4 = 4 * atan (/5) - atan(/239).
Lemma Machin_2_3_7 : PI/4 = 2 * atan(/3) + (atan (/7)).