Hint Rewrite gather_prop_left gather_prop_right : gather_prop.

(* Specifications: *)
Lemma andp_in_order1 {A}{NA: NatDed A}:
forall P Q, P && Q = P && (P --> Q).
Lemma andp_in_order2 {A}{NA: NatDed A}:
forall P Q, P && Q = Q && (Q --> P).
Lemma andp_right1{A}{NA: NatDed A}:
forall P Q R, P |-- Q -> P && Q |-- R -> P |-- Q && R.
Lemma andp_right2{A}{NA: NatDed A}:
forall P Q R, P |-- R -> P && R |-- Q -> P |-- Q && R.