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(* 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.

Theorem Rlt_pow_R1 :
forall (e : R) (n : nat), (1 < e)%R -> 0 < n -> (1 < e ^ n)%R.