Use the Scheme command to generate the maximal induction principle for le.
The following text defines a function with a pre-condition based on le and proves a small theorem about this function.
Definition pred_partial: forall (n : nat), n <> 0 -> nat. intros n; case n. intros h; elim h; reflexivity. intros p h'; exact p. Defined. Theorem le_2_n_not_zero: forall (n : nat), 2 <= n -> (n <> 0). Proof. intros n Hle; elim Hle; intros; discriminate. Qed.
Prove the following theorem, using the maximal induction principle.
Theorem le_2_n_pred: forall (n : nat) (h : 2 <= n), (pred_partial n (le_2_n_not_zero n h) <> 0).