(** Definition of order for quadroples and a binary sequnce *)


Record Z_pos : Set :=  {zposcrr :> Z; z4prf_Z_pos : (0 <= zposcrr)%Z}.(** Definition of order for quadroples and a binary sequnce *)


Record Z_pos : Set :=  {zposcrr :> Z; z4prf_Z_pos : (0 <= zposcrr)%Z}.(** Definition of order for quadroples and a binary sequnce *)


Record Z_pos : Set :=  {zposcrr :> Z; z4prf_Z_pos : (0 <= zposcrr)%Z}.Definition qlt (a b c d : Z) (p : Qpositive) (a' b' c' d' : Z)
  (p' : Qpositive) : Prop :=
  bin_lt p p' \/ p = p' /\ (a + b + c + d < a' + b' + c' + d')%Z.Definition qle (a b c d : Z) (p : Qpositive) (a' b' c' d' : Z)
  (p' : Qpositive) : Prop :=
  qlt a b c d p a' b' c' d' p' \/
  a = a' /\ b = b' /\ c = c' /\ d = d' /\ p = p'.Definition quadrointegral_lt (a b c d a' b' c' d' : Z) :=
  (a + b + c + d < a' + b' + c' + d')%Z.Definition quadrointegral_eq (a b c d a' b' c' d' : Z) :=
  a = a' /\ b = b' /\ c = c' /\ d = d'.Record Z4 : Set := 
  {z4crr :> Z * Z * (Z * Z);
   z4prf :
    (0 <= fst (fst z4crr))%Z /\
    (0 <= snd (fst z4crr))%Z /\
    (0 <= fst (snd z4crr))%Z /\ (0 <= snd (snd z4crr))%Z}.Record Z4 : Set := 
  {z4crr :> Z * Z * (Z * Z);
   z4prf :
    (0 <= fst (fst z4crr))%Z /\
    (0 <= snd (fst z4crr))%Z /\
    (0 <= fst (snd z4crr))%Z /\ (0 <= snd (snd z4crr))%Z}.Record Z4 : Set := 
  {z4crr :> Z * Z * (Z * Z);
   z4prf :
    (0 <= fst (fst z4crr))%Z /\
    (0 <= snd (fst z4crr))%Z /\
    (0 <= fst (snd z4crr))%Z /\ (0 <= snd (snd z4crr))%Z}.Definition Z4_lt (x y : Z4) :=
  let (V1, V2) := z4crr x in
  let (V3, V4) := z4crr y in
  let (a, b) := V1 in
  let (c, d) := V2 in
  let (a', b') := V3 in
  let (c', d') := V4 in quadrointegral_lt a b c d a' b' c' d'.Definition Z4_eq (x y : Z4) :=
  let (V1, V2) := z4crr x in
  let (V3, V4) := z4crr y in
  let (a, b) := V1 in
  let (c, d) := V2 in
  let (a', b') := V3 in
  let (c', d') := V4 in quadrointegral_eq a b c d a' b' c' d'.

Lemma Z4_lt_is_transitive : forall x y z : Z4, Z4_lt x y -> Z4_lt y z -> Z4_lt x z.