Definition rel_prod : relation (A * B) :=
    fun p1 p2 => (relA (fst p1) (fst p2) /\ relB (snd p1) (snd p2)).Global Instance prod_proper : Proper (relA ==> relB ==> rel_prod) (@pair A B).Global Instance equiv_prod : Equivalence rel_prod.Instance SepAlgOps_prod: SepAlgOps (A * B) :=
  { sa_unit ab := sa_unit (fst ab) /\ sa_unit (snd ab)
  ; sa_mul a b c :=
      sa_mul (fst a) (fst b) (fst c) /\
      sa_mul (snd a) (snd b) (snd c)
  }.Definition SepAlg_prod: SepAlg (rel := rel_prod (relA := rel) (relB := rel0)) (A * B).

Lemma sa_mul_split (a b c : A) (a' b' c' : B) : sa_mul (a, a') (b, b') (c, c') <-> sa_mul a b c /\ sa_mul a' b' c'.