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(** Properties of subtraction between natural numbers.  This file is mostly OBSOLETE now, see module [PeanoNat.Nat] instead.  [minus] is now an alias for [Nat.sub], which is defined in [Init/Nat.v] as: << Fixpoint sub (n m:nat) : nat := match n, m with | S k, S l => k - l | _, _ => n end where "n - m" := (sub n m) : nat_scope. >> *)
Local Open Scope nat_scope.
(** * 0 is right neutral *)
(** * Permutation with successor *)
Register pred_of_minus as num.nat.pred_of_minus.
(** * Diagonal *)
Notation minus_diag := Nat.sub_diag (only parsing). (* n - n = 0 *)
Notation minus_n_n := minus_diag_reverse.
(** * Simplification *)

Lemma minus_plus_simpl_l_reverse n m p : n - m = p + n - (p + m).