theorem le_to_or_lt_eq: ∀n,m:nat. n ≤ m → n < m ∨ n = m.
#n #m #lenm (elim lenm) /3/ qed.
+theorem eq_or_gt: ∀n. 0 = n ∨ 0 < n.
+#n elim (le_to_or_lt_eq 0 n ?) // /2 width=1/
+qed-.
+
theorem increasing_to_le2: ∀f:nat → nat. increasing f →
∀m:nat. f 0 ≤ m → ∃i. f i ≤ m ∧ m < f (S i).
#f #incr #m #lem (elim lem)