diff --git a/algebra/upred.v b/algebra/upred.v
index 3910a66c6902472c8fbd5b40275010de1631e1f1..f2a8a55e548fa13af8b63a7882ca378d0f066d54 100644
--- a/algebra/upred.v
+++ b/algebra/upred.v
@@ -615,6 +615,10 @@ Lemma sep_intro_True_l P Q R : True ⊑ P → R ⊑ Q → R ⊑ (P ★ Q).
 Proof. by intros; rewrite -(left_id True%I uPred_sep R); apply sep_mono. Qed.
 Lemma sep_intro_True_r P Q R : R ⊑ P → True ⊑ Q → R ⊑ (P ★ Q).
 Proof. by intros; rewrite -(right_id True%I uPred_sep R); apply sep_mono. Qed.
+Lemma sep_elim_True_l P Q R : True ⊑ P → (P ★ R) ⊑ Q → R ⊑ Q.
+Proof. by intros HP; rewrite -HP left_id. Qed.
+Lemma sep_elim_True_r P Q R : True ⊑ P → (R ★ P) ⊑ Q → R ⊑ Q.
+Proof. by intros HP; rewrite -HP right_id. Qed.
 Lemma wand_intro_l P Q R : (Q ★ P) ⊑ R → P ⊑ (Q -★ R).
 Proof. rewrite comm; apply wand_intro_r. Qed.
 Lemma wand_elim_r P Q : (P ★ (P -★ Q)) ⊑ Q.
diff --git a/heap_lang/heap.v b/heap_lang/heap.v
index 5a85ee3b6f4b7d372db492c1bf564ab14cef82ec..3f589d43f2f07ccabff1d6ce3f5e033b050bea6c 100644
--- a/heap_lang/heap.v
+++ b/heap_lang/heap.v
@@ -119,8 +119,10 @@ Section heap.
     P ⊑ (▷ (∀ l, heap_mapsto HeapI γ l v -★ Q (LocV l))) →
     P ⊑ wp E (Alloc e) Q.
   Proof.
-    intros HN ? Hctx HP. rewrite -(right_id True%I (★)%I (P)) (auth_empty γ) pvs_frame_l.
-    apply wp_strip_pvs. eapply wp_alloc_heap with (σ:=∅); try eassumption.
+    intros HN ? Hctx HP. eapply sep_elim_True_r.
+    { eapply (auth_empty γ). }
+    rewrite pvs_frame_l. apply wp_strip_pvs.
+    eapply wp_alloc_heap with (σ:=∅); try eassumption.
     { eauto with I. }
     rewrite HP comm. apply sep_mono.
     - rewrite /heap_own. f_equiv. apply: map_eq=>l. by rewrite lookup_fmap !lookup_empty.
@@ -166,7 +168,8 @@ Section heap.
     P ⊑ wp E (Store (Loc l) e) Q.
   Proof.
     rewrite /heap_ctx /heap_own. intros Hl Hval HN Hctx HP.
-    eapply (auth_fsa (heap_inv HeapI) (wp_fsa _) (λ _:(), alter (λ _, Excl v) l)); simpl; eauto.
+    eapply (auth_fsa (heap_inv HeapI) (wp_fsa _)
+                     (λ _:(), alter (λ _, Excl v) l)); simpl; eauto.
     rewrite HP=>{HP Hctx HN}. apply sep_mono; first done.
     apply forall_intro=>hf. apply wand_intro_l. rewrite /heap_inv.
     rewrite -assoc. apply const_elim_sep_l=>Hv /=.
@@ -189,12 +192,13 @@ Section heap.
       + by rewrite lookup_alter /to_heap !lookup_fmap lookup_insert Hl /=.
       + rewrite lookup_alter_ne // !lookup_fmap lookup_insert_ne //. }
     rewrite !EQ. apply sep_mono; last done.
-    f_equiv. apply: map_eq=>l'. move: (Hv 0%nat l'). destruct (decide (l=l')); simplify_map_equality.
+    f_equiv. apply: map_eq=>l'. move: (Hv 0%nat l').
+    destruct (decide (l=l')); simplify_map_equality.
     - rewrite /from_heap /to_heap lookup_insert lookup_omap !lookup_op.
       rewrite !lookup_fmap lookup_insert Hl.
       case (hf !! l')=>[[?||]|]; auto; contradiction.
-    - rewrite /from_heap /to_heap lookup_insert_ne // !lookup_omap !lookup_op !lookup_fmap.
-      rewrite lookup_insert_ne //.
+    - rewrite /from_heap /to_heap lookup_insert_ne // !lookup_omap.
+      rewrite !lookup_op !lookup_fmap lookup_insert_ne //.
   Qed.
 
   Lemma wp_store N E γ l v' e v P Q :
@@ -204,7 +208,8 @@ Section heap.
     P ⊑ (heap_mapsto HeapI γ l v' ★ ▷ (heap_mapsto HeapI γ l v -★ Q (LitV LitUnit))) →
     P ⊑ wp E (Store (Loc l) e) Q.
   Proof.
-    rewrite /heap_mapsto=>Hval HN Hctx HP. eapply wp_store_heap; try eassumption; last first.
+    rewrite /heap_mapsto=>Hval HN Hctx HP.
+    eapply wp_store_heap; try eassumption; last first.
     - rewrite HP. apply sep_mono; first done. by rewrite insert_singleton.
     - by rewrite lookup_insert.
   Qed.
diff --git a/program_logic/auth.v b/program_logic/auth.v
index cb5f4b8245201aac5a9bb6741ed05e4b2ede85f1..31e35703742df120fc56846e1d0eccb3d3ffb53d 100644
--- a/program_logic/auth.v
+++ b/program_logic/auth.v
@@ -32,8 +32,8 @@ Section auth.
   Lemma auth_alloc N a :
     ✓ a → φ a ⊑ pvs N N (∃ γ, auth_ctx AuthI γ N φ ∧ auth_own AuthI γ a).
   Proof.
-    intros Ha. rewrite -(right_id True%I (★)%I (φ _)).
-    rewrite (own_alloc AuthI (Auth (Excl a) a) N) //; [].
+    intros Ha. eapply sep_elim_True_r.
+    { by eapply (own_alloc AuthI (Auth (Excl a) a) N). }
     rewrite pvs_frame_l. apply pvs_strip_pvs.
     rewrite sep_exist_l. apply exist_elim=>γ. rewrite -(exist_intro γ).
     transitivity (▷ auth_inv AuthI γ φ ★ auth_own AuthI γ a)%I.