proposition: Lemma pow_incr : forall (x y:R) (n:nat), 0 <= x <= y -> x ^ n <= y ^ n.
proof: Proof. intros. induction n as [| n Hrecn]. - right; reflexivity. - simpl. elim H; intros. apply Rle_trans with (y * x ^ n). + do 2 rewrite <- (Rmult_comm (x ^ n)). apply Rmult_le_compat_l. * apply pow_le; assumption. * assumption. + apply Rmult_le_compat_l. * apply Rle_trans with x; assumption. * apply Hrecn. Qed.
imports: ArithRing Rdefinitions Raxioms RIneq Rpow_def R_Ifp Rbasic_fun R_sqr SplitAbsolu SplitRmult ArithProp Zpower Ztac