End NatSeq.
Notation seq := seq.
(***********************)
(** ** List comparison *)
(***********************)
Section Compare.
Variable A : Type.
Variable cmp : A -> A -> comparison.
Local Notation list_compare := (@list_compare A cmp).
Section Lemmas.
Variable Hcmp : forall x y, cmp x y = Eq <-> x = y.

Lemma list_compare_cons (x : A) (xs ys : list A) :
list_compare (x :: xs) (x :: ys) = list_compare xs ys.