Ltac reduceR :=
match goal with
| |- _ |-- |==> |> FF || _ => apply aux1_reduceR
| _ => idtac
end;
match goal with
| |- _ |-- |==> _ => apply aux2_reduceR
| _ => idtac
end.

Lemma aux_reduceL: forall P Q R S: environ -> mpred,
P && R |-- S ->
P && (Q && R) |-- S.