+
+theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
+(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
+P (leb n m).
+apply nat_elim2; intros; simplify
+ [apply H.apply le_O_n
+ |apply H1.apply not_le_Sn_O.
+ |apply H;intros
+ [apply H1.apply le_S_S.assumption.
+ |apply H2.unfold Not.intros.apply H3.apply le_S_S_to_le.assumption
+ ]
+ ]
+qed.
+
+(*
+theorem decidable_le: \forall n,m. n \leq m \lor n \nleq m.
+intros.
+apply (leb_elim n m)
+ [intro.left.assumption
+ |intro.right.assumption
+ ]
+qed.
+*)
+
+theorem le_to_leb_true: \forall n,m. n \leq m \to leb n m = true.
+intros.apply leb_elim;intros
+ [reflexivity
+ |apply False_ind.apply H1.apply H.
+ ]
+qed.
+
+theorem lt_to_leb_false: \forall n,m. m < n \to leb n m = false.
+intros.apply leb_elim;intros
+ [apply False_ind.apply (le_to_not_lt ? ? H1). assumption
+ |reflexivity
+ ]
+qed.
+