Section SeqOrd.
Variable (T : ordType).
Fixpoint ords x  : pred (seq T) :=
fun y => match x , y with
| [::] , [::] => false
| [::] ,  t :: ts => true
| x :: xs , y :: ys => if x == y then ords xs ys
else ord x y
| _ :: _ , [::] => false
end.

Lemma total_ords : forall x y, [|| ords x y, x == y | ords y x].