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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** Properties of addition.  This file is mostly OBSOLETE now, see module [PeanoNat.Nat] instead.  [Nat.add] is defined in [Init/Nat.v] as: << Fixpoint add (n m:nat) : nat := match n with | O => m | S p => S (p + m) end where "n + m" := (add n m) : nat_scope. >> *)
Local Open Scope nat_scope.
(** * Neutrality of 0, commutativity, associativity *)
Notation plus_0_l := Nat.add_0_l (only parsing).
Notation plus_0_r := Nat.add_0_r (only parsing).
Notation plus_comm := Nat.add_comm (only parsing).
Notation plus_assoc := Nat.add_assoc (only parsing).
Notation plus_permute := Nat.add_shuffle3 (only parsing).
Definition plus_Snm_nSm : forall n m, S n + m = n + S m :=
Peano.plus_n_Sm.
(** * Simplification *)
(** * Compatibility with order *)

Lemma plus_lt_le_compat n m p q : n < m -> p <= q -> n + p < m + q.