End NthTheory.

(* Specifications: *)
Lemma set_nth_default T s (y0 x0 : T) n : n < size s -> nth x0 s n = nth y0 s n.
Lemma size_incr_nth v i :
size (incr_nth v i) = if i < size v then size v else i.+1.
Lemma incr_nth_inj v : injective (incr_nth v).
Lemma incr_nthC v i j :
incr_nth (incr_nth v i) j = incr_nth (incr_nth v j) i.