(** *** Proof that tangent is a bijection *)
(* to be removed? *)

(* Specifications: *)
Lemma derive_increasing_interv : forall (a b : R) (f : R -> R),
a < b ->
forall (pr:forall x, a < x < b -> derivable_pt f x),
(forall t:R, forall (t_encad : a < t < b), 0 < derive_pt f t (pr t t_encad)) ->
forall x y:R, a < x < b -> a < y < b -> x < y -> f x < f y.
(* begin hide *)
Lemma plus_Rsqr_gt_0 : forall x, 1 + x ^ 2 > 0.
(* end hide *)
(* The following lemmas about PI should probably be in Rtrigo. *)
Lemma PI2_lower_bound : forall x, 0 < x < 2 -> 0 < cos x -> x < PI/2.
Lemma PI2_3_2 : 3/2 < PI/2.
Lemma PI2_1 : 1 < PI/2.
Lemma tan_inj : forall x y,
-PI/2 < x < PI/2 -> -PI/2 < y < PI/2 ->
tan x = tan y ->
x = y.