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.
End Center.
Proof (order_induction 0).
Proof (order_induction' 0).
(** Elimintation principle for < *)

Theorem lt_ind : forall (n : t),
A (S n) ->
(forall m, n < m -> A m -> A (S m)) ->
forall m, n < m -> A m.