proposition: Theorem nle_gt : ∀ n m, ¬n ≤ m ↔ m < n.
proof: Proof. intros n m. split; intros H. destruct n as [n| ]. destruct m as [m| ]; [ idtac | exfalso; apply H; constructor ]. apply fin_lt_mono, Nat.nlt_ge. intros I; apply H. apply fin_le_mono, le_S_n; assumption. destruct m; [ constructor | exfalso; apply H; constructor ]. destruct n as [n| ]. destruct m as [m| ]; [ idtac | inversion H ]. apply fin_lt_mono, Nat.nle_gt in H. intros I; apply H; clear H. apply fin_le_mono; assumption. destruct m; [ intros I; inversion I | inversion H ]. Qed.
imports: Utf8 Compare_dec NPeano Misc