(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(****************************************************************************  IEEE754  :  Rpower  Laurent Thery  ***************************************************************************** Definition of an exponential function over relative numbers *)
(* We have already an exponential over natural number, we prove some basic properties for this function *)
simpl in |- *; auto with real.
Qed.
simpl in |- *; auto with real.
Qed.
intros e n; elim n; simpl in |- *; auto with real.
Qed.
intros e n; elim n; simpl in |- *; auto with real.
intros n0 H' m; rewrite H'; auto with real.
Qed.
Hint Resolve pow_O pow_1 pow_NR0 pow_add: real.
intros e n; elim n; simpl in |- *; auto with real.
intros n0 H' m H'0.
rewrite Rmult_assoc; rewrite <- H'; auto.
Qed.
intros e n; elim n; simpl in |- *; auto with real.
intros n0 H' H'0; replace 0%R with (e * 0)%R; auto with real.
Qed.
Hint Resolve pow_lt: real.
intros e n; elim n; simpl in |- *; auto with real.
intros H' H'0; Contradict H'0; auto with arith.
intros n0; case n0.
simpl in |- *; rewrite Rmult_1_r; auto.
intros n1 H' H'0 H'1.
replace 1%R with (1 * 1)%R; auto with real.
apply Rlt_trans with (r2 := (e * 1)%R); auto with real.
apply Rmult_lt_compat_l; auto with real.
apply Rlt_trans with (r2 := 1%R); auto with real.
apply H'; auto with arith.
Qed.
Hint Resolve Rlt_pow_R1: real.

Theorem Rlt_pow :
forall (e : R) (n m : nat), (1 < e)%R -> n < m -> (e ^ n < e ^ m)%R.