Section AllRel.
Variables (T S : Type) (r : T -> S -> bool).
Implicit Types (x : T) (y : S) (xs : seq T) (ys : seq S).
Definition allrel xs ys := all [pred x | all (r x) ys] xs.

Lemma sub_allrel {T S : Type} (r r' : T -> S -> bool) :
(forall x y, r x y -> r' x y) ->
forall xs ys, allrel r xs ys -> allrel r' xs ys.