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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) *)
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(*    //   *    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.
(** This file is similar to [PermutSetoid], except that the equality used here is Coq usual one instead of a setoid equality. In particular, we can then prove the equivalence between [Permutation.Permutation] and [PermutSetoid.permutation]. *)
Section Perm.
Variable A : Type.
Hypothesis eq_dec : forall x y:A, {x=y} + {~ x=y}.
Notation permutation := (permutation _ eq_dec).
Notation list_contents := (list_contents _ eq_dec).
(** we can use [multiplicity] to define [In] and [NoDup]. *)

Lemma multiplicity_In_O :
forall l a, ~ In a l -> multiplicity (list_contents l) a = 0.