Section lb_mask_nat.
Let lb_mask_nat_1 l : ⟬ ⟦l⟧⟭  ⪯  l.
Let lb_mask_nat_2 l : l ⪯  ⟬ ⟦l⟧⟭  .
Fact lb_mask_nat l : ⟬ ⟦l⟧⟭ ≂ l.
Hint Resolve lb_mask_eq_binary_le binary_le_eq_lb_mask : core.
Fact binary_le_trans x y z : x ≲ y -> y ≲ z -> x ≲ z.
Fact lb_mask_equiv_equal l m : l ≂ m <-> ⟦l⟧ = ⟦m⟧.
Fact equal_lb_mask_equiv x y : x = y <-> ⟬ x⟭ ≂⟬ y⟭.
Local Notation lbeq := lb_mask_equiv (only parsing).
Add Parametric Morphism: (lb_nat) with signature (lbeq) ==> (eq) as lb_nat_eq.
End binary_le_dio.
Definition lb_dio (R : (nat -> lb) -> Prop) := { f | forall v, df_pred f (fun n => ⟦v n⟧) <-> R v }.

Theorem lb_mask_diophantine x y : lb_dio (fun v => v x ⪯  v y).