Record big_rel_class_of T (leq : rel T) :=
  BigRelClass {
      leq_big_internal_op : rel T;
      bigger_than_op : seq T -> T;
      _ : leq_big_internal_op = leq;
      _ : forall i s, leq_big_internal_op i (bigger_than_op (i :: s));
      _ : forall i j s, leq_big_internal_op i (bigger_than_op s) ->
                        leq_big_internal_op i (bigger_than_op (j :: s))
}.Record big_rel_class_of T (leq : rel T) :=
  BigRelClass {
      leq_big_internal_op : rel T;
      bigger_than_op : seq T -> T;
      _ : leq_big_internal_op = leq;
      _ : forall i s, leq_big_internal_op i (bigger_than_op (i :: s));
      _ : forall i j s, leq_big_internal_op i (bigger_than_op s) ->
                        leq_big_internal_op i (bigger_than_op (j :: s))
}.Record big_rel_class_of T (leq : rel T) :=
  BigRelClass {
      leq_big_internal_op : rel T;
      bigger_than_op : seq T -> T;
      _ : leq_big_internal_op = leq;
      _ : forall i s, leq_big_internal_op i (bigger_than_op (i :: s));
      _ : forall i j s, leq_big_internal_op i (bigger_than_op s) ->
                        leq_big_internal_op i (bigger_than_op (j :: s))
}.Record big_rel_class_of T (leq : rel T) :=
  BigRelClass {
      leq_big_internal_op : rel T;
      bigger_than_op : seq T -> T;
      _ : leq_big_internal_op = leq;
      _ : forall i s, leq_big_internal_op i (bigger_than_op (i :: s));
      _ : forall i j s, leq_big_internal_op i (bigger_than_op s) ->
                        leq_big_internal_op i (bigger_than_op (j :: s))
}.Record big_rel_of T := BigRelOf {
  leq_big :> rel T;
  big_rel_class : big_rel_class_of leq_big
}.Record big_rel_of T := BigRelOf {
  leq_big :> rel T;
  big_rel_class : big_rel_class_of leq_big
}.Record big_rel_of T := BigRelOf {
  leq_big :> rel T;
  big_rel_class : big_rel_class_of leq_big
}.Record big_rel_of T := BigRelOf {
  leq_big :> rel T;
  big_rel_class : big_rel_class_of leq_big
}.Definition bigger_than_of T (b : big_rel_of T)
           (phb : phantom (rel T) b) :=
  bigger_than_op (big_rel_class b).Notation bigger_than leq := (@bigger_than_of _ _ (Phantom (rel _) leq)).Definition leq_big_internal_of T (b : big_rel_of T)
           (phb : phantom (rel T) b) :=
  leq_big_internal_op (big_rel_class b).Notation leq_big_internal leq := (@leq_big_internal_of _ _ (Phantom (rel _) leq)).

Lemma next_bigger_than T (b : big_rel_of T) i j s : leq_big_internal b i (bigger_than b s) -> leq_big_internal b i (bigger_than b (j :: s)).