(************************************************************************)
(*         *      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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Set Implicit Arguments.
Module RingSyntax.
Reserved Notation "x ?=! y" (at level 70, no associativity).
Reserved Notation "x +! y " (at level 50, left associativity).
Reserved Notation "x -! y" (at level 50, left associativity).
Reserved Notation "x *! y" (at level 40, left associativity).
Reserved Notation "-! x" (at level 35, right associativity).
Reserved Notation "[ x ]" (at level 0).
Reserved Notation "x ?== y" (at level 70, no associativity).
Reserved Notation "x -- y" (at level 50, left associativity).
Reserved Notation "x ** y" (at level 40, left associativity).
Reserved Notation "-- x" (at level 35, right associativity).
Reserved Notation "x == y" (at level 70, no associativity).
End RingSyntax.
(* Set Universe Polymorphism. *)
Section Power.
Variable R:Type.
Variable rI : R.
Variable rmul : R -> R -> R.
Variable req : R -> R -> Prop.
Variable Rsth : Equivalence req.
Infix "*" := rmul.
Infix "==" := req.
Hypothesis mul_ext : Proper (req ==> req ==> req) rmul.
Hypothesis mul_assoc  : forall x y z, x * (y * z) == (x * y) * z.
Fixpoint pow_pos (x:R) (i:positive) : R :=
match i with
| xH => x
| xO i => let p := pow_pos x i in p * p
| xI i => let p := pow_pos x i in x * (p * p)
end.

Lemma pow_pos_swap x j : pow_pos x j * x == x * pow_pos x j.