Section LeftInduction.
Let A' (n : t) := forall m, m <= z -> n <= m -> A m.
Let left_step :=   forall n, n < z -> A (S n) -> A n.
Let left_step' :=  forall n, n <= z -> A' (S n) -> A n.
Let left_step'' := forall n, A' n <-> A' (S n).
End LeftInduction.

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.