prove lemma in both directions

theories/fin_maps.v

@@ -1234,9 +1234,15 @@ Proof.
 Qed.
 Lemma map_Forall_delete m i : map_Forall P m → map_Forall P (delete i m).
 Proof. intros Hm j x; rewrite lookup_delete_Some. naive_solver. Qed.
-Lemma map_Forall_lookup m i x :
-  m !! i = Some x → map_Forall P m → P i x.
-Proof. intros Hlk Hm. by apply Hm. Qed.
+Lemma map_Forall_lookup m :
+  map_Forall P m ↔ ∀ i x, m !! i = Some x → P i x.
+Proof. done. Qed.
+Lemma map_Forall_lookup_1 m i x :
+  map_Forall P m → m !! i = Some x → P i x.
+Proof. intros ?. by apply map_Forall_lookup. Qed.
+Lemma map_Forall_lookup_2 m :
+  (∀ i x, m !! i = Some x → P i x) → map_Forall P m.
+Proof. intros ?. by apply map_Forall_lookup. Qed.
 Lemma map_Forall_foldr_delete m is :
   map_Forall P m → map_Forall P (foldr delete m is).
 Proof. induction is; eauto using map_Forall_delete. Qed.