Definition Qpositive2 := (Qpositive * Qpositive)%type.
Definition bin2_lt (x y : Qpositive2) :=
let (p1, p2) := x in let (p1', p2) := y in bin_lt p1 p1'.
Definition bin2_eq (x y : Qpositive2) :=
let (p1, p2) := x in let (p1', p2) := y in p1 = p1'.

Lemma octet_lt_wf_rec_without_zeros_and_One :
forall
P : Z -> Z -> Z -> Z -> Z -> Z -> Z -> Z -> Qpositive -> Qpositive -> Prop,
(forall (a b c d e f g h : Z_pos) (q1 q2 : Qpositive),
(forall (k l r s t u v w : Z_pos) (p1 p2 : Qpositive),
octet_lt k l r s t u v w p1 p2 a b c d e f g h q1 q2 ->
P k l r s t u v w p1 p2) -> P a b c d e f g h q1 q2) ->
forall (a b c d e f g h : Z_pos) (q1 q2 : Qpositive),
P a b c d e f g h q1 q2.