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(** * Definition of hyperbolic functions *)
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Definition sinh (x:R) : R := (exp x - exp (- x)) / 2.
Definition tanh (x:R) : R := sinh x / cosh x.

(* Specifications: *)
Lemma cosh_0 : cosh 0 = 1.
Lemma sinh_0 : sinh 0 = 0.
Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)).
Lemma simpl_cos_n :
forall n:nat, cos_n (S n) / cos_n n = - / INR (2 * S n * (2 * n + 1)).
Lemma archimed_cor1 :
forall eps:R, 0 < eps ->  exists N : nat, / INR N < eps /\ (0 < N)%nat.
Lemma Alembert_cos : Un_cv (fun n:nat => Rabs (cos_n (S n) / cos_n n)) 0.
Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0.
Definition cos_in (x l:R) : Prop :=
infinite_sum (fun i:nat => cos_n i * x ^ i) l.