Definition dec_predicate (m : Message) :=
  match m with
  | dec => true
  | _ => false
  end.Definition inc_predicate (m : Message) :=
  match m with
  | inc_dec _ => true
  | _ => false
  end.Definition site_inc_predicate (m : Message) :=
  match m with
  | inc_dec s => if eq_site_dec s s0 then true else false
  | _ => false
  end.Definition copy_predicate (m : Message) :=
  match m with
  | copy => true
  | _ => false
  end.Definition dec_count (m : Message) :=
  match m with
  | dec => 1%Z
  | _ => 0%Z
  end.Definition inc_count (m : Message) := 0%Z.Definition copy_count (m : Message) :=
  match m with
  | copy => 1%Z
  | _ => 0%Z
  end.Definition cardinal_count :=
  fun_sum Message dec_count (fun_sum Message inc_count copy_count).Definition cardinal := reduce Message cardinal_count.

Lemma disjoint_cardinal : forall q : queue Message, cardinal q = (reduce Message dec_count q + (reduce Message inc_count q + reduce Message copy_count q))%Z.