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 eq_in_allrel {T S : Type} (P : {pred T}) (Q : {pred S}) r r' :
{in P & Q, r =2 r'} ->
forall xs ys, all P xs -> all Q ys -> allrel r xs ys = allrel r' xs ys.