Get rid of some `uPred M` coercions.

theories/algebra/agree.v

@@ -233,13 +233,13 @@ Lemma agree_op_invL' `{!LeibnizEquiv A} a b : ✓ (to_agree a ⋅ to_agree b)
 Proof. by intros ?%agree_op_inv'%leibniz_equiv. Qed.
 
 (** Internalized properties *)
-Lemma agree_equivI {M} a b : to_agree a ≡ to_agree b ⊣⊢ (a ≡ b : uPred M).
+Lemma agree_equivI {M} a b : to_agree a ≡ to_agree b ⊣⊢@{uPredI M} (a ≡ b).
 Proof.
   uPred.unseal. do 2 split.
   - intros Hx. exact: to_agree_injN.
   - intros Hx. exact: to_agree_ne.
 Qed.
-Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊢ (x ≡ y : uPred M).
+Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊢@{uPredI M} x ≡ y.
 Proof. uPred.unseal; split=> r n _ ?; by apply: agree_op_invN. Qed.
 End agree.
 

theories/algebra/auth.v

@@ -191,15 +191,15 @@ Global Instance auth_frag_core_id a : CoreId a → CoreId (◯ a).
 Proof. do 2 constructor; simpl; auto. by apply core_id_core. Qed.
 
 (** Internalized properties *)
-Lemma auth_equivI {M} (x y : auth A) :
-  x ≡ y ⊣⊢ (authoritative x ≡ authoritative y ∧ auth_own x ≡ auth_own y : uPred M).
+Lemma auth_equivI {M} x y :
+  x ≡ y ⊣⊢@{uPredI M} authoritative x ≡ authoritative y ∧ auth_own x ≡ auth_own y.
 Proof. by uPred.unseal. Qed.
-Lemma auth_validI {M} (x : auth A) :
-  ✓ x ⊣⊢ (match authoritative x with
-          | Excl' a => (∃ b, a ≡ auth_own x ⋅ b) ∧ ✓ a
-          | None => ✓ auth_own x
-          | ExclBot' => False
-          end : uPred M).
+Lemma auth_validI {M} x :
+  ✓ x ⊣⊢@{uPredI M} match authoritative x with
+                    | Excl' a => (∃ b, a ≡ auth_own x ⋅ b) ∧ ✓ a
+                    | None => ✓ auth_own x
+                    | ExclBot' => False
+                    end.
 Proof. uPred.unseal. by destruct x as [[[]|]]. Qed.
 
 Lemma auth_frag_op a b : ◯ (a ⋅ b) = ◯ a ⋅ ◯ b.

theories/algebra/csum.v

@@ -284,22 +284,22 @@ Proof. intros ? [] ? EQ; inversion_clear EQ. by eapply id_free0_r. Qed.
 
 (** Internalized properties *)
 Lemma csum_equivI {M} (x y : csum A B) :
-  x ≡ y ⊣⊢ (match x, y with
-            | Cinl a, Cinl a' => a ≡ a'
-            | Cinr b, Cinr b' => b ≡ b'
-            | CsumBot, CsumBot => True
-            | _, _ => False
-            end : uPred M).
+  x ≡ y ⊣⊢@{uPredI M} match x, y with
+                      | Cinl a, Cinl a' => a ≡ a'
+                      | Cinr b, Cinr b' => b ≡ b'
+                      | CsumBot, CsumBot => True
+                      | _, _ => False
+                      end.
 Proof.
   uPred.unseal; do 2 split; first by destruct 1.
   by destruct x, y; try destruct 1; try constructor.
 Qed.
 Lemma csum_validI {M} (x : csum A B) :
-  ✓ x ⊣⊢ (match x with
-          | Cinl a => ✓ a
-          | Cinr b => ✓ b
-          | CsumBot => False
-          end : uPred M).
+  ✓ x ⊣⊢@{uPredI M} match x with
+                    | Cinl a => ✓ a
+                    | Cinr b => ✓ b
+                    | CsumBot => False
+                    end.
 Proof. uPred.unseal. by destruct x. Qed.
 
 (** Updates *)

theories/algebra/excl.v

@@ -95,17 +95,17 @@ Global Instance excl_cmra_discrete : OfeDiscrete A → CmraDiscrete exclR.
 Proof. split. apply _. by intros []. Qed.
 
 (** Internalized properties *)
-Lemma excl_equivI {M} (x y : excl A) :
-  x ≡ y ⊣⊢ (match x, y with
-            | Excl a, Excl b => a ≡ b
-            | ExclBot, ExclBot => True
-            | _, _ => False
-            end : uPred M).
+Lemma excl_equivI {M} x y :
+  x ≡ y ⊣⊢@{uPredI M} match x, y with
+                      | Excl a, Excl b => a ≡ b
+                      | ExclBot, ExclBot => True
+                      | _, _ => False
+                      end.
 Proof.
   uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor.
 Qed.
-Lemma excl_validI {M} (x : excl A) :
-  ✓ x ⊣⊢ (if x is ExclBot then False else True : uPred M).
+Lemma excl_validI {M} x :
+  ✓ x ⊣⊢@{uPredI M} if x is ExclBot then False else True.
 Proof. uPred.unseal. by destruct x. Qed.
 
 (** Exclusive *)

theories/algebra/gmap.v

@@ -177,9 +177,9 @@ Qed.
 Canonical Structure gmapUR := UcmraT (gmap K A) gmap_ucmra_mixin.
 
 (** Internalized properties *)
-Lemma gmap_equivI {M} m1 m2 : m1 ≡ m2 ⊣⊢ (∀ i, m1 !! i ≡ m2 !! i : uPred M).
+Lemma gmap_equivI {M} m1 m2 : m1 ≡ m2 ⊣⊢@{uPredI M} ∀ i, m1 !! i ≡ m2 !! i.
 Proof. by uPred.unseal. Qed.
-Lemma gmap_validI {M} m : ✓ m ⊣⊢ (∀ i, ✓ (m !! i) : uPred M).
+Lemma gmap_validI {M} m : ✓ m ⊣⊢@{uPredI M} ∀ i, ✓ (m !! i).
 Proof. by uPred.unseal. Qed.
 End cmra.
 

theories/algebra/list.v

@@ -247,9 +247,9 @@ Section cmra.
   Qed.
 
   (** Internalized properties *)
-  Lemma list_equivI {M} l1 l2 : l1 ≡ l2 ⊣⊢ (∀ i, l1 !! i ≡ l2 !! i : uPred M).
+  Lemma list_equivI {M} l1 l2 : l1 ≡ l2 ⊣⊢@{uPredI M} ∀ i, l1 !! i ≡ l2 !! i.
   Proof. uPred.unseal; constructor=> n x ?. apply list_dist_lookup. Qed.
-  Lemma list_validI {M} l : ✓ l ⊣⊢ (∀ i, ✓ (l !! i) : uPred M).
+  Lemma list_validI {M} l : ✓ l ⊣⊢@{uPredI M} ∀ i, ✓ (l !! i).
   Proof. uPred.unseal; constructor=> n x ?. apply list_lookup_validN. Qed.
 End cmra.