+theorem f_min_true: \forall f:nat \to bool. \forall m:nat.
+(\exists i. le i m \land f i = true) \to
+f (min m f) = true.
+intros.unfold min.
+apply f_min_aux_true.
+elim H.clear H.elim H1.clear H1.
+apply (ex_intro ? ? a).
+split
+ [split
+ [apply le_O_n
+ |rewrite < plus_n_O.assumption
+ ]
+ |assumption
+ ]
+qed.
+