Local Open Scope logic.
Hint Rewrite <- prop_and : gather_prop.

(* Specifications: *)
Lemma gather_prop_left {A}{NA: NatDed A}:
forall P Q R,  !! P && (!! Q && R) = !!(P/\Q) && R.
Lemma gather_prop_right {A}{NA: NatDed A}:
forall P Q R,  R && !! P && !! Q = !!(P/\Q) && R.