(************************************************************************)
(*         *      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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(** Well-founded relations and natural numbers *)
Local Open Scope nat_scope.
Implicit Types m n p : nat.
Section Well_founded_Nat.
Variable A : Type.
Variable f : A -> nat.
Definition ltof (a b:A) := f a < f b.
Definition gtof (a b:A) := f b > f a.

(* Specifications: *)
Theorem well_founded_ltof : well_founded ltof.
Lemma gt_wf_rect :
forall n (P:nat -> Type), (forall n, (forall m, n > m -> P m) -> P n) -> P n.