End LeftInduction.

(* Specifications: *)
Theorem order_induction :
A z ->
(forall n, z <= n -> A n -> A (S n)) ->
(forall n, n < z  -> A (S n) -> A n) ->
forall n, A n.
Theorem order_induction' :
A z ->
(forall n, z <= n -> A n -> A (S n)) ->
(forall n, n <= z -> A n -> A (P n)) ->
forall n, A n.