Section Between.
Variables P Q : nat -> Prop.
(** The [between] type expresses the concept [forall i: nat, k <= i < l -> P i.]. *)
Inductive between k : nat -> Prop :=
| bet_emp : between k k
| bet_S : forall l, between k l -> P l -> between k (S l).
Hint Constructors between: arith.
Hint Resolve bet_eq: arith.
Hint Immediate between_le: arith.
Hint Resolve between_Sk_l: arith.

Lemma between_restr :
forall k l (m:nat), k <= l -> l <= m -> between k m -> between l m.