+lemma weak_pad1 :∀n,a.∃b. n ≤ 〈a,b〉.
+#n #a
+cut (∀i.decidable (〈a,i〉 < n))
+ [#i @decidable_le ]
+ #Hdec cases(decidable_forall (λb. 〈a,b〉 < n) Hdec n)
+ [#H cut (∀i. i < n → ∃b. b < n ∧ 〈a,b〉 = i)
+ [@(injective_to_exists … H) //]
+ #Hcut %{n} @not_lt_to_le % #Han
+ lapply(Hcut ? Han) * #x * #Hx #Hx2
+ cut (x = n) [//] #Hxn >Hxn in Hx; /2 by absurd/
+ |#H lapply(not_forall_to_exists … Hdec H)
+ * #b * #H1 #H2 %{b} @not_lt_to_le @H2
+ ]
+qed.
+
+lemma pad : ∀n,a. ∃b. n ≤ |〈a,b〉|.
+#n #a cases (weak_pad1 (of_size n) a) #b #Hb
+%{b} <(size_of_size n) @monotonic_size //
+qed.