(* The right step lemma is just the transitivity of eq *)
Declare Right Step (@Equivalence_Transitive _ _ eq_equiv).

(* Specifications: *)
Theorem succ_inj : forall n1 n2, S n1 == S n2 -> n1 == n2.
(* The following theorem is useful as an equivalence for proving bidirectional induction steps *)
Theorem succ_inj_wd : forall n1 n2, S n1 == S n2 <-> n1 == n2.
Theorem succ_inj_wd_neg : forall n m, S n ~= S m <-> n ~= m.
Section CentralInduction.