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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** Properties of the power function *)
(** Derived properties of power, specialized on natural numbers *)
Module Type NPowProp
(Import A : NAxiomsSig')
(Import B : NSubProp A)
(Import C : NParityProp A B).
Module Import Private_NZPow := Nop <+ NZPowProp A A B.
Ltac auto' := trivial; try rewrite <- neq_0_lt_0; auto using le_0_l.
Ltac wrap l := intros; apply l; auto'.

(* Specifications: *)
Lemma pow_succ_r' : forall a b, a^(S b) == a * a^b.
(** Monotonicity *)
Lemma pow_lt_mono_l : forall a b c, c~=0 -> a<b -> a^c < b^c.
(** For any a>1, the a^x function is above the identity function *)
Lemma pow_gt_lin_r : forall a b, 1<a -> b < a^b.
Lemma odd_pow : forall a b, b~=0 -> odd (a^b) = odd a.
End NPowProp.