simplify.reflexivity.
simplify.apply not_eq_O_S.
intro.
-simplify.
+simplify.unfold Not.
intro. apply (not_eq_O_S n1).apply sym_eq.assumption.
intros.simplify.
generalize in match H.
elim ((eqb n1 m1)).
simplify.apply eq_f.apply H1.
-simplify.intro.apply H1.apply inj_S.assumption.
+simplify.unfold Not.intro.apply H1.apply inj_S.assumption.
qed.
theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
simplify.exact not_le_Sn_O.
intros 2.simplify.elim ((leb n1 m1)).
simplify.apply le_S_S.apply H.
-simplify.intros.apply H.apply le_S_S_to_le.assumption.
+simplify.unfold Not.intros.apply H.apply le_S_S_to_le.assumption.
qed.
theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
| EQ \Rightarrow n=m
| GT \Rightarrow m < n ])).
intro.elim n1.simplify.reflexivity.
-simplify.apply le_S_S.apply le_O_n.
-intro.simplify.apply le_S_S. apply le_O_n.
+simplify.unfold lt.apply le_S_S.apply le_O_n.
+intro.simplify.unfold lt.apply le_S_S. apply le_O_n.
intros 2.simplify.elim ((nat_compare n1 m1)).
-simplify. apply le_S_S.apply H.
+simplify. unfold lt. apply le_S_S.apply H.
simplify. apply eq_f. apply H.
-simplify. apply le_S_S.apply H.
+simplify. unfold lt.apply le_S_S.apply H.
qed.
theorem nat_compare_n_m_m_n: \forall n,m:nat.