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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope R_scope.
(* 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.
(* A simple formula, reasonably efficient. *)
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)).
Definition PI_2_3_7_tg n :=
2 * Ratan_seq (/3) n + Ratan_seq (/7) n.