(* define nonnegative dyadic rationals *)
Inductive dyadic_rational : Type :=
  | m_over_2_to_n: forall m n:nat, dyadic_rational.(* define nonnegative dyadic rationals *)
Inductive dyadic_rational : Type :=
  | m_over_2_to_n: forall m n:nat, dyadic_rational.Inductive dr_eq : dyadic_rational -> dyadic_rational -> Prop :=
  | dr_eq_refl: forall x:dyadic_rational, dr_eq x x
  | dr_eq_sym: forall x y:dyadic_rational, dr_eq x y -> dr_eq y x
  | dr_eq_trans: forall x y z:dyadic_rational,
    dr_eq x y -> dr_eq y z -> dr_eq x z
  | dr_eq_scale: forall (m n:nat),
    dr_eq (m_over_2_to_n m n)
          (m_over_2_to_n (2*m) (S n)).Inductive dr_eq : dyadic_rational -> dyadic_rational -> Prop :=
  | dr_eq_refl: forall x:dyadic_rational, dr_eq x x
  | dr_eq_sym: forall x y:dyadic_rational, dr_eq x y -> dr_eq y x
  | dr_eq_trans: forall x y z:dyadic_rational,
    dr_eq x y -> dr_eq y z -> dr_eq x z
  | dr_eq_scale: forall (m n:nat),
    dr_eq (m_over_2_to_n m n)
          (m_over_2_to_n (2*m) (S n)).Inductive dr_eq : dyadic_rational -> dyadic_rational -> Prop :=
  | dr_eq_refl: forall x:dyadic_rational, dr_eq x x
  | dr_eq_sym: forall x y:dyadic_rational, dr_eq x y -> dr_eq y x
  | dr_eq_trans: forall x y z:dyadic_rational,
    dr_eq x y -> dr_eq y z -> dr_eq x z
  | dr_eq_scale: forall (m n:nat),
    dr_eq (m_over_2_to_n m n)
          (m_over_2_to_n (2*m) (S n)).Inductive dr_eq : dyadic_rational -> dyadic_rational -> Prop :=
  | dr_eq_refl: forall x:dyadic_rational, dr_eq x x
  | dr_eq_sym: forall x y:dyadic_rational, dr_eq x y -> dr_eq y x
  | dr_eq_trans: forall x y z:dyadic_rational,
    dr_eq x y -> dr_eq y z -> dr_eq x z
  | dr_eq_scale: forall (m n:nat),
    dr_eq (m_over_2_to_n m n)
          (m_over_2_to_n (2*m) (S n)).Inductive dr_eq : dyadic_rational -> dyadic_rational -> Prop :=
  | dr_eq_refl: forall x:dyadic_rational, dr_eq x x
  | dr_eq_sym: forall x y:dyadic_rational, dr_eq x y -> dr_eq y x
  | dr_eq_trans: forall x y z:dyadic_rational,
    dr_eq x y -> dr_eq y z -> dr_eq x z
  | dr_eq_scale: forall (m n:nat),
    dr_eq (m_over_2_to_n m n)
          (m_over_2_to_n (2*m) (S n)).Inductive dr_lt : dyadic_rational -> dyadic_rational -> Prop :=
  | intro_dr_lt: forall m m' n:nat, m < m' ->
    dr_lt (m_over_2_to_n m n) (m_over_2_to_n m' n)
  | dr_lt_wd: forall x x' y y':dyadic_rational,
    dr_lt x y -> dr_eq x x' -> dr_eq y y' -> dr_lt x' y'.Inductive dr_lt : dyadic_rational -> dyadic_rational -> Prop :=
  | intro_dr_lt: forall m m' n:nat, m < m' ->
    dr_lt (m_over_2_to_n m n) (m_over_2_to_n m' n)
  | dr_lt_wd: forall x x' y y':dyadic_rational,
    dr_lt x y -> dr_eq x x' -> dr_eq y y' -> dr_lt x' y'.Inductive dr_lt : dyadic_rational -> dyadic_rational -> Prop :=
  | intro_dr_lt: forall m m' n:nat, m < m' ->
    dr_lt (m_over_2_to_n m n) (m_over_2_to_n m' n)
  | dr_lt_wd: forall x x' y y':dyadic_rational,
    dr_lt x y -> dr_eq x x' -> dr_eq y y' -> dr_lt x' y'.

Lemma dr_incr_denom: forall (m n n':nat), n<=n' -> exists m':nat, dr_eq (m_over_2_to_n m' n') (m_over_2_to_n m n).