(** * Shifting an interval by some amount *)
Definition shift (i: interv) (delta: Z) : interv := (fst i + delta, snd i + delta).

(* Specifications: *)
Lemma in_shift:
forall x i delta,
In x i -> In (x + delta) (shift i delta).
Lemma in_shift_inv:
forall x i delta,
In x (shift i delta) -> In (x - delta) i.
Lemma in_shift:
forall x i delta,
In x i -> In (x + delta) (shift i delta).
Lemma in_shift_inv:
forall x i delta,
In x (shift i delta) -> In (x - delta) i.