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).
*)
Definition bool_add_with_rem a b c : bool * bool :=
match a, b, c with
| ⟘, ⟘, ⟘ => (⟘,⟘)
| ⟘, ⟘, ⟙ => (⟘,⟙)
| ⟘, ⟙, ⟘ => (⟘,⟙)
| ⟘, ⟙, ⟙ => (⟙,⟘)
| ⟙, ⟘, ⟘ => (⟘,⟙)
| ⟙, ⟘, ⟙ => (⟙,⟘)
| ⟙, ⟙, ⟘ => (⟙,⟘)
| ⟙, ⟙, ⟙ => (⟙,⟙)
end.