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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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(** * Functions on finite domains *)
(** Main result : for functions [f:A->A] with finite [A], f injective <-> f bijective <-> f surjective. *)
#[local] Set Warnings "-stdlib-vector".
Set Implicit Arguments.
(** General definitions *)
Definition Injective {A B} (f : A->B) :=
forall x y, f x = f y -> x = y.
Definition Surjective {A B} (f : A->B) :=
forall y, exists x, f x = y.
Definition Bijective {A B} (f : A->B) :=
exists g:B->A, (forall x, g (f x) = x) /\ (forall y, f (g y) = y).
(** Finiteness is defined here via exhaustive list enumeration *)
Definition Full {A:Type} (l:list A) := forall a:A, In a l.
Definition Finite (A:Type) := exists (l:list A), Full l.
(** In many of the following proofs, it will be convenient to have list enumerations without duplicates. As soon as we have decidability of equality (in Prop), this is equivalent to the previous notion (s. lemma Finite_dec). *)
Definition Listing {A:Type} (l:list A) := NoDup l /\ Full l.
Definition Finite' (A:Type) := exists (l:list A), Listing l.

Lemma Finite_dec {A:Type}: Finite A /\ decidable_eq A <-> Finite' A.