End RightInduction.
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).

(* Specifications: *)
Theorem strong_left_induction: left_step' -> forall n, n <= z -> A n.
Theorem left_induction : A z -> left_step -> forall n, n <= z -> A n.
Theorem left_induction' :
(forall n, z <= n -> A n) -> left_step -> forall n, A n.
Theorem strong_left_induction' :
(forall n, z <= n -> A n) -> left_step' -> forall n, A n.