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(** Equality is decidable on [nat] *)
Local Open Scope nat_scope.
Notation not_eq_sym := not_eq_sym (only parsing).
Implicit Types m n p q : nat.
Definition le_or_le_S := le_le_S_dec.
Definition Pcompare := gt_eq_gt_dec.
Proof le_ge_dec.
Definition lt_or_eq n m := {m > n} + {n = m}.
Proof le_lt_eq_dec.

Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m.