(* arcsinh function *)

Definition arcsinh x := ln (x + sqrt (x ^ 2 + 1)).

(* Specifications: *)
Lemma arcsinh_sinh : forall x, arcsinh (sinh x) = x.
Lemma sinh_arcsinh x : sinh (arcsinh x) = x.
Lemma derivable_pt_lim_arcsinh : forall x, derivable_pt_lim arcsinh x (/sqrt (x ^ 2 + 1)).
Lemma arcsinh_lt : forall x y, x < y -> arcsinh x < arcsinh y.
Lemma arcsinh_le : forall x y, x <= y -> arcsinh x <= arcsinh y.
Lemma arcsinh_0 : arcsinh 0 = 0.