(************************************************************************)
(*         *      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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(** Power Function *)
(** Interface of a power function, then its specification on naturals *)
Module Type Pow (Import A : Typ).
Parameters Inline pow : t -> t -> t.
End Pow.
Module Type PowNotation (A : Typ)(Import B : Pow A).
Infix "^" := pow.
End PowNotation.
Module Type Pow' (A : Typ) := Pow A <+ PowNotation A.
Module Type NZPowSpec (Import A : NZOrdAxiomsSig')(Import B : Pow' A).
#[global]
Declare Instance pow_wd : Proper (eq==>eq==>eq) pow.

(* Specifications: *)
Axiom pow_0_r : forall a, a^0 == 1.
Axiom pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b.
Axiom pow_neg_r : forall a b, b<0 -> a^b == 0.
End NZPowSpec.