make entailment notation look like entailment

algebra/agree.v

@@ -133,11 +133,11 @@ Lemma to_agree_car n (x : agree A) : ✓{n} x → to_agree (x n) ≡{n}≡ x.
 Proof. intros [??]; split; naive_solver eauto using agree_valid_le. Qed.
 
 (** Internalized properties *)
-Lemma agree_equivI {M} a b : (to_agree a ≡ to_agree b)%I ≡ (a ≡ b : uPred M)%I.
+Lemma agree_equivI {M} a b : (to_agree a ≡ to_agree b) ⊣⊢ (a ≡ b : uPred M).
 Proof.
   uPred.unseal. do 2 split. by intros [? Hv]; apply (Hv n). apply: to_agree_ne.
 Qed.
-Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊑ (x ≡ y : uPred M).
+Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊢ (x ≡ y : uPred M).
 Proof. uPred.unseal; split=> r n _ ?; by apply: agree_op_inv. Qed.
 End agree.
 

algebra/auth.v

@@ -144,14 +144,14 @@ Qed.
 
 (** Internalized properties *)
 Lemma auth_equivI {M} (x y : auth A) :
-  (x ≡ y)%I ≡ (authoritative x ≡ authoritative y ∧ own x ≡ own y : uPred M)%I.
+  (x ≡ y) ⊣⊢ (authoritative x ≡ authoritative y ∧ own x ≡ own y : uPred M).
 Proof. by uPred.unseal. Qed.
 Lemma auth_validI {M} (x : auth A) :
-  (✓ x)%I ≡ (match authoritative x with
+  (✓ x) ⊣⊢ (match authoritative x with
              | Excl a => (∃ b, a ≡ own x ⋅ b) ∧ ✓ a
              | ExclUnit => ✓ own x
              | ExclBot => False
-             end : uPred M)%I.
+             end : uPred M).
 Proof. uPred.unseal. by destruct x as [[]]. Qed.
 
 (** The notations ◯ and ● only work for CMRAs with an empty element. So, in

algebra/excl.v

@@ -145,16 +145,16 @@ Qed.
 
 (** Internalized properties *)
 Lemma excl_equivI {M} (x y : excl A) :
-  (x ≡ y)%I ≡ (match x, y with
+  (x ≡ y) ⊣⊢ (match x, y with
                | Excl a, Excl b => a ≡ b
                | ExclUnit, ExclUnit | ExclBot, ExclBot => True
                | _, _ => False
-               end : uPred M)%I.
+               end : uPred M).
 Proof.
   uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor.
 Qed.
 Lemma excl_validI {M} (x : excl A) :
-  (✓ x)%I ≡ (if x is ExclBot then False else True : uPred M)%I.
+  (✓ x) ⊣⊢ (if x is ExclBot then False else True : uPred M).
 Proof. uPred.unseal. by destruct x. Qed.
 
 (** ** Local updates *)

algebra/fin_maps.v

@@ -170,9 +170,9 @@ Global Instance map_cmra_discrete : CMRADiscrete A → CMRADiscrete mapR.
 Proof. split; [apply _|]. intros m ? i. by apply: cmra_discrete_valid. Qed.
 
 (** Internalized properties *)
-Lemma map_equivI {M} m1 m2 : (m1 ≡ m2)%I ≡ (∀ i, m1 !! i ≡ m2 !! i : uPred M)%I.
+Lemma map_equivI {M} m1 m2 : (m1 ≡ m2) ⊣⊢ (∀ i, m1 !! i ≡ m2 !! i : uPred M).
 Proof. by uPred.unseal. Qed.
-Lemma map_validI {M} m : (✓ m)%I ≡ (∀ i, ✓ (m !! i) : uPred M)%I.
+Lemma map_validI {M} m : (✓ m) ⊣⊢ (∀ i, ✓ (m !! i) : uPred M).
 Proof. by uPred.unseal. Qed.
 End cmra.
 

algebra/frac.v

@@ -190,17 +190,17 @@ Proof. intros. by apply frac_validN_inv_l with 0 a, cmra_valid_validN. Qed.
 
 (** Internalized properties *)
 Lemma frac_equivI {M} (x y : frac A) :
-  (x ≡ y)%I ≡ (match x, y with
+  (x ≡ y) ⊣⊢ (match x, y with
                | Frac q1 a, Frac q2 b => q1 = q2 ∧ a ≡ b
                | FracUnit, FracUnit => True
                | _, _ => False
-               end : uPred M)%I.
+               end : uPred M).
 Proof.
   uPred.unseal; do 2 split; first by destruct 1.
   by destruct x, y; destruct 1; try constructor.
 Qed.
 Lemma frac_validI {M} (x : frac A) :
-  (✓ x)%I ≡ (if x is Frac q a then ■ (q ≤ 1)%Qc ∧ ✓ a else True : uPred M)%I.
+  (✓ x) ⊣⊢ (if x is Frac q a then ■ (q ≤ 1)%Qc ∧ ✓ a else True : uPred M).
 Proof. uPred.unseal. by destruct x. Qed.
 
 (** ** Local updates *)

algebra/iprod.v

@@ -170,9 +170,9 @@ Section iprod_cmra.
   Qed.
 
   (** Internalized properties *)
-  Lemma iprod_equivI {M} g1 g2 : (g1 ≡ g2)%I ≡ (∀ i, g1 i ≡ g2 i : uPred M)%I.
+  Lemma iprod_equivI {M} g1 g2 : (g1 ≡ g2) ⊣⊢ (∀ i, g1 i ≡ g2 i : uPred M).
   Proof. by uPred.unseal. Qed.
-  Lemma iprod_validI {M} g : (✓ g)%I ≡ (∀ i, ✓ g i : uPred M)%I.
+  Lemma iprod_validI {M} g : (✓ g) ⊣⊢ (∀ i, ✓ g i : uPred M).
   Proof. by uPred.unseal. Qed.
 
   (** Properties of iprod_insert. *)

algebra/option.v

@@ -138,14 +138,14 @@ Proof. by destruct mx, my; inversion_clear 1. Qed.
 
 (** Internalized properties *)
 Lemma option_equivI {M} (x y : option A) :
-  (x ≡ y)%I ≡ (match x, y with
+  (x ≡ y) ⊣⊢ (match x, y with
                | Some a, Some b => a ≡ b | None, None => True | _, _ => False
-               end : uPred M)%I.
+               end : uPred M).
 Proof.
   uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor.
 Qed.
 Lemma option_validI {M} (x : option A) :
-  (✓ x)%I ≡ (match x with Some a => ✓ a | None => True end : uPred M)%I.
+  (✓ x) ⊣⊢ (match x with Some a => ✓ a | None => True end : uPred M).
 Proof. uPred.unseal. by destruct x. Qed.
 
 (** Updates *)

algebra/upred_big_op.v

@@ -39,9 +39,9 @@ Implicit Types Ps Qs : list (uPred M).
 Implicit Types A : Type.
 
 (* Big ops *)
-Global Instance big_and_proper : Proper ((≡) ==> (≡)) (@uPred_big_and M).
+Global Instance big_and_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_and M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
-Global Instance big_sep_proper : Proper ((≡) ==> (≡)) (@uPred_big_sep M).
+Global Instance big_sep_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_sep M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 
 Global Instance big_and_ne n :
@@ -51,19 +51,19 @@ Global Instance big_sep_ne n :
   Proper (Forall2 (dist n) ==> dist n) (@uPred_big_sep M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 
-Global Instance big_and_mono' : Proper (Forall2 (⊑) ==> (⊑)) (@uPred_big_and M).
+Global Instance big_and_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_and M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
-Global Instance big_sep_mono' : Proper (Forall2 (⊑) ==> (⊑)) (@uPred_big_sep M).
+Global Instance big_sep_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_sep M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 
-Global Instance big_and_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_and M).
+Global Instance big_and_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_and M).
 Proof.
   induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
   - by rewrite IH.
   - by rewrite !assoc (comm _ P).
   - etrans; eauto.
 Qed.
-Global Instance big_sep_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_sep M).
+Global Instance big_sep_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_sep M).
 Proof.
   induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
   - by rewrite IH.
@@ -71,26 +71,26 @@ Proof.
   - etrans; eauto.
 Qed.
 
-Lemma big_and_app Ps Qs : (Π∧ (Ps ++ Qs))%I ≡ (Π∧ Ps ∧ Π∧ Qs)%I.
+Lemma big_and_app Ps Qs : (Π∧ (Ps ++ Qs)) ⊣⊢ (Π∧ Ps ∧ Π∧ Qs).
 Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
-Lemma big_sep_app Ps Qs : (Π★ (Ps ++ Qs))%I ≡ (Π★ Ps ★ Π★ Qs)%I.
+Lemma big_sep_app Ps Qs : (Π★ (Ps ++ Qs)) ⊣⊢ (Π★ Ps ★ Π★ Qs).
 Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
 
-Lemma big_and_contains Ps Qs : Qs `contains` Ps → (Π∧ Ps)%I ⊑ (Π∧ Qs)%I.
+Lemma big_and_contains Ps Qs : Qs `contains` Ps → (Π∧ Ps) ⊢ (Π∧ Qs).
 Proof.
   intros [Ps' ->]%contains_Permutation. by rewrite big_and_app and_elim_l.
 Qed.
-Lemma big_sep_contains Ps Qs : Qs `contains` Ps → (Π★ Ps)%I ⊑ (Π★ Qs)%I.
+Lemma big_sep_contains Ps Qs : Qs `contains` Ps → (Π★ Ps) ⊢ (Π★ Qs).
 Proof.
   intros [Ps' ->]%contains_Permutation. by rewrite big_sep_app sep_elim_l.
 Qed.
 
-Lemma big_sep_and Ps : (Π★ Ps) ⊑ (Π∧ Ps).
+Lemma big_sep_and Ps : (Π★ Ps) ⊢ (Π∧ Ps).
 Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed.
 
-Lemma big_and_elem_of Ps P : P ∈ Ps → (Π∧ Ps) ⊑ P.
+Lemma big_and_elem_of Ps P : P ∈ Ps → (Π∧ Ps) ⊢ P.
 Proof. induction 1; simpl; auto with I. Qed.
-Lemma big_sep_elem_of Ps P : P ∈ Ps → (Π★ Ps) ⊑ P.
+Lemma big_sep_elem_of Ps P : P ∈ Ps → (Π★ Ps) ⊢ P.
 Proof. induction 1; simpl; auto with I. Qed.
 
 (* Big ops over finite maps *)
@@ -100,8 +100,8 @@ Section gmap.
   Implicit Types Φ Ψ : K → A → uPred M.
 
   Lemma big_sepM_mono Φ Ψ m1 m2 :
-    m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → Φ k x ⊑ Ψ k x) →
-    (Π★{map m1} Φ) ⊑ (Π★{map m2} Ψ).
+    m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → Φ k x ⊢ Ψ k x) →
+    (Π★{map m1} Φ) ⊢ (Π★{map m2} Ψ).
   Proof.
     intros HX HΦ. trans (Π★{map m2} Φ)%I.
     - by apply big_sep_contains, fmap_contains, map_to_list_contains.
@@ -117,36 +117,36 @@ Section gmap.
     apply Forall2_Forall, Forall_true=> -[i x]; apply HΦ.
   Qed.
   Global Instance big_sepM_proper m :
-    Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡))
+    Proper (pointwise_relation _ (pointwise_relation _ (⊣⊢)) ==> (⊣⊢))
            (uPred_big_sepM (M:=M) m).
   Proof.
     intros Φ1 Φ2 HΦ; apply equiv_dist=> n.
     apply big_sepM_ne=> k x; apply equiv_dist, HΦ.
   Qed.
   Global Instance big_sepM_mono' m :
-    Proper (pointwise_relation _ (pointwise_relation _ (⊑)) ==> (⊑))
+    Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (⊢))
            (uPred_big_sepM (M:=M) m).
   Proof. intros Φ1 Φ2 HΦ. apply big_sepM_mono; intros; [done|apply HΦ]. Qed.
 
-  Lemma big_sepM_empty Φ : (Π★{map ∅} Φ)%I ≡ True%I.
+  Lemma big_sepM_empty Φ : (Π★{map ∅} Φ) ⊣⊢ True.
   Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed.
   Lemma big_sepM_insert Φ (m : gmap K A) i x :
-    m !! i = None → (Π★{map <[i:=x]> m} Φ)%I ≡ (Φ i x ★ Π★{map m} Φ)%I.
+    m !! i = None → (Π★{map <[i:=x]> m} Φ) ⊣⊢ (Φ i x ★ Π★{map m} Φ).
   Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed.
-  Lemma big_sepM_singleton Φ i x : (Π★{map {[i := x]}} Φ)%I ≡ (Φ i x)%I.
+  Lemma big_sepM_singleton Φ i x : (Π★{map {[i := x]}} Φ) ⊣⊢ (Φ i x).
   Proof.
     rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty.
     by rewrite big_sepM_empty right_id.
   Qed.
 
   Lemma big_sepM_sepM Φ Ψ m :
-    (Π★{map m} (λ i x, Φ i x ★ Ψ i x))%I ≡ (Π★{map m} Φ ★ Π★{map m} Ψ)%I.
+    (Π★{map m} (λ i x, Φ i x ★ Ψ i x)) ⊣⊢ (Π★{map m} Φ ★ Π★{map m} Ψ).
   Proof.
     rewrite /uPred_big_sepM.
     induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //.
     by rewrite IH -!assoc (assoc _ (Ψ _ _)) [(Ψ _ _ ★ _)%I]comm -!assoc.
   Qed.
-  Lemma big_sepM_later Φ m : (▷ Π★{map m} Φ)%I ≡ (Π★{map m} (λ i x, ▷ Φ i x))%I.
+  Lemma big_sepM_later Φ m : (▷ Π★{map m} Φ) ⊣⊢ (Π★{map m} (λ i x, ▷ Φ i x)).
   Proof.
     rewrite /uPred_big_sepM.
     induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?later_True //.
@@ -161,7 +161,7 @@ Section gset.
   Implicit Types Φ : A → uPred M.
 
   Lemma big_sepS_mono Φ Ψ X Y :
-    Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊑ Ψ x) → (Π★{set X} Φ) ⊑ (Π★{set Y} Ψ).
+    Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊢ Ψ x) → (Π★{set X} Φ) ⊢ (Π★{set Y} Ψ).
   Proof.
     intros HX HΦ. trans (Π★{set Y} Φ)%I.
     - by apply big_sep_contains, fmap_contains, elements_contains.
@@ -176,38 +176,38 @@ Section gset.
     apply Forall2_Forall, Forall_true=> x; apply HΦ.
   Qed.
   Lemma big_sepS_proper X :
-    Proper (pointwise_relation _ (≡) ==> (≡)) (uPred_big_sepS (M:=M) X).
+    Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (uPred_big_sepS (M:=M) X).
   Proof.
     intros Φ1 Φ2 HΦ; apply equiv_dist=> n.
     apply big_sepS_ne=> x; apply equiv_dist, HΦ.
   Qed.
   Lemma big_sepS_mono' X :
-    Proper (pointwise_relation _ (⊑) ==> (⊑)) (uPred_big_sepS (M:=M) X).
+    Proper (pointwise_relation _ (⊢) ==> (⊢)) (uPred_big_sepS (M:=M) X).
   Proof. intros Φ1 Φ2 HΦ. apply big_sepS_mono; naive_solver. Qed.
 
-  Lemma big_sepS_empty Φ : (Π★{set ∅} Φ)%I ≡ True%I.
+  Lemma big_sepS_empty Φ : (Π★{set ∅} Φ) ⊣⊢ True.
   Proof. by rewrite /uPred_big_sepS elements_empty. Qed.
   Lemma big_sepS_insert Φ X x :
-    x ∉ X → (Π★{set {[ x ]} ∪ X} Φ)%I ≡ (Φ x ★ Π★{set X} Φ)%I.
+    x ∉ X → (Π★{set {[ x ]} ∪ X} Φ) ⊣⊢ (Φ x ★ Π★{set X} Φ).
   Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed.
   Lemma big_sepS_delete Φ X x :
-    x ∈ X → (Π★{set X} Φ)%I ≡ (Φ x ★ Π★{set X ∖ {[ x ]}} Φ)%I.
+    x ∈ X → (Π★{set X} Φ) ⊣⊢ (Φ x ★ Π★{set X ∖ {[ x ]}} Φ).
   Proof.
     intros. rewrite -big_sepS_insert; last set_solver.
     by rewrite -union_difference_L; last set_solver.
   Qed.
-  Lemma big_sepS_singleton Φ x : (Π★{set {[ x ]}} Φ)%I ≡ (Φ x)%I.
+  Lemma big_sepS_singleton Φ x : (Π★{set {[ x ]}} Φ) ⊣⊢ (Φ x).
   Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed.
 
   Lemma big_sepS_sepS Φ Ψ X :
-    (Π★{set X} (λ x, Φ x ★ Ψ x))%I ≡ (Π★{set X} Φ ★ Π★{set X} Ψ)%I.
+    (Π★{set X} (λ x, Φ x ★ Ψ x)) ⊣⊢ (Π★{set X} Φ ★ Π★{set X} Ψ).
   Proof.
     rewrite /uPred_big_sepS.
     induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id.
     by rewrite IH -!assoc (assoc _ (Ψ _)) [(Ψ _ ★ _)%I]comm -!assoc.
   Qed.
 
-  Lemma big_sepS_later Φ X : (▷ Π★{set X} Φ)%I ≡ (Π★{set X} (λ x, ▷ Φ x))%I.
+  Lemma big_sepS_later Φ X : (▷ Π★{set X} Φ) ⊣⊢ (Π★{set X} (λ x, ▷ Φ x)).
   Proof.
     rewrite /uPred_big_sepS.
     induction (elements X) as [|x l IH]; csimpl; first by rewrite ?later_True.

algebra/upred_tactics.v

@@ -24,18 +24,18 @@ Module uPred_reflection. Section uPred_reflection.
 
   Notation eval_list Σ l :=
     (uPred_big_sep ((λ n, from_option True%I (Σ !! n)) <$> l)).
-  Lemma eval_flatten Σ e : eval Σ e ≡ eval_list Σ (flatten e).
+  Lemma eval_flatten Σ e : eval Σ e ⊣⊢ eval_list Σ (flatten e).
   Proof.
     induction e as [| |e1 IH1 e2 IH2];
       rewrite /= ?right_id ?fmap_app ?big_sep_app ?IH1 ?IH2 //. 
   Qed.
   Lemma flatten_entails Σ e1 e2 :
-    flatten e2 `contains` flatten e1 → eval Σ e1 ⊑ eval Σ e2.
+    flatten e2 `contains` flatten e1 → eval Σ e1 ⊢ eval Σ e2.
   Proof.
     intros. rewrite !eval_flatten. by apply big_sep_contains, fmap_contains.
   Qed.
   Lemma flatten_equiv Σ e1 e2 :
-    flatten e2 ≡ₚ flatten e1 → eval Σ e1 ≡ eval Σ e2.
+    flatten e2 ≡ₚ flatten e1 → eval Σ e1 ⊣⊢ eval Σ e2.
   Proof. intros He. by rewrite !eval_flatten He. Qed.
 
   Fixpoint prune (e : expr) : expr :=
@@ -54,7 +54,7 @@ Module uPred_reflection. Section uPred_reflection.
     induction e as [| |e1 IH1 e2 IH2]; simplify_eq/=; auto.
     rewrite -IH1 -IH2. by repeat case_match; rewrite ?right_id_L.
   Qed.
-  Lemma prune_correct Σ e : eval Σ (prune e) ≡ eval Σ e.
+  Lemma prune_correct Σ e : eval Σ (prune e) ⊣⊢ eval Σ e.
   Proof. by rewrite !eval_flatten flatten_prune. Qed.
 
   Fixpoint cancel_go (n : nat) (e : expr) : option expr :=
@@ -86,7 +86,7 @@ Module uPred_reflection. Section uPred_reflection.
   Qed.
   Lemma cancel_entails Σ e1 e2 e1' e2' ns :
     cancel ns e1 = Some e1' → cancel ns e2 = Some e2' →
-    eval Σ e1' ⊑ eval Σ e2' → eval Σ e1 ⊑ eval Σ e2.
+    eval Σ e1' ⊢ eval Σ e2' → eval Σ e1 ⊢ eval Σ e2.
   Proof.
     intros ??. rewrite !eval_flatten.
     rewrite (flatten_cancel e1 e1' ns) // (flatten_cancel e2 e2' ns) //; csimpl.
@@ -100,20 +100,20 @@ Module uPred_reflection. Section uPred_reflection.
     | n :: l => ESep (EVar n) (to_expr l)
     end.
   Arguments to_expr !_ / : simpl nomatch.
-  Lemma eval_to_expr Σ l : eval Σ (to_expr l) ≡ eval_list Σ l.
+  Lemma eval_to_expr Σ l : eval Σ (to_expr l) ⊣⊢ eval_list Σ l.
   Proof.
     induction l as [|n1 [|n2 l] IH]; csimpl; rewrite ?right_id //.
     by rewrite IH.
   Qed.
 
   Lemma split_l Σ e ns e' :
-    cancel ns e = Some e' → eval Σ e ≡ (eval Σ (to_expr ns) ★ eval Σ e')%I.
+    cancel ns e = Some e' → eval Σ e ⊣⊢ (eval Σ (to_expr ns) ★ eval Σ e').
   Proof.
     intros He%flatten_cancel.
     by rewrite eval_flatten He fmap_app big_sep_app eval_to_expr eval_flatten.
   Qed.
   Lemma split_r Σ e ns e' :
-    cancel ns e = Some e' → eval Σ e ≡ (eval Σ e' ★ eval Σ (to_expr ns))%I.
+    cancel ns e = Some e' → eval Σ e ⊣⊢ (eval Σ e' ★ eval Σ (to_expr ns)).
   Proof. intros. rewrite /= comm. by apply split_l. Qed.
 
   Class Quote (Σ1 Σ2 : list (uPred M)) (P : uPred M) (e : expr) := {}.
@@ -132,16 +132,16 @@ Module uPred_reflection. Section uPred_reflection.
 
   Ltac quote :=
     match goal with
-    | |- ?P1 ⊑ ?P2 =>
+    | |- ?P1 ⊢ ?P2 =>
       lazymatch type of (_ : Quote [] _ P1 _) with Quote _ ?Σ2 _ ?e1 =>
       lazymatch type of (_ : Quote Σ2 _ P2 _) with Quote _ ?Σ3 _ ?e2 =>
-        change (eval Σ3 e1 ⊑ eval Σ3 e2) end end
+        change (eval Σ3 e1 ⊢ eval Σ3 e2) end end
     end.
   Ltac quote_l :=
     match goal with
-    | |- ?P1 ⊑ ?P2 =>
+    | |- ?P1 ⊢ ?P2 =>
       lazymatch type of (_ : Quote [] _ P1 _) with Quote _ ?Σ2 _ ?e1 =>
-        change (eval Σ2 e1 ⊑ P2) end
+        change (eval Σ2 e1 ⊢ P2) end
     end.
 End uPred_reflection.
 
@@ -162,7 +162,7 @@ Ltac close_uPreds Ps tac :=
 
 Tactic Notation "cancel" constr(Ps) :=
   uPred_reflection.quote;
-  let Σ := match goal with |- uPred_reflection.eval ?Σ _ ⊑ _ => Σ end in
+  let Σ := match goal with |- uPred_reflection.eval ?Σ _ ⊢ _ => Σ end in
   let ns' := lazymatch type of (_ : uPred_reflection.QuoteArgs Σ Ps _) with
              | uPred_reflection.QuoteArgs _ _ ?ns' => ns'
              end in
@@ -172,12 +172,12 @@ Tactic Notation "cancel" constr(Ps) :=
 Tactic Notation "ecancel" open_constr(Ps) :=
   close_uPreds Ps ltac:(fun Qs => cancel Qs).
 
-(** [to_front [P1, P2, ..]] rewrites in the premise of ⊑ such that
+(** [to_front [P1, P2, ..]] rewrites in the premise of ⊢ such that
     the assumptions P1, P2, ... appear at the front, in that order. *)
 Tactic Notation "to_front" open_constr(Ps) :=
   close_uPreds Ps ltac:(fun Ps =>
     uPred_reflection.quote_l;
-    let Σ := match goal with |- uPred_reflection.eval ?Σ _ ⊑ _ => Σ end in
+    let Σ := match goal with |- uPred_reflection.eval ?Σ _ ⊢ _ => Σ end in
     let ns' := lazymatch type of (_ : uPred_reflection.QuoteArgs Σ Ps _) with
                | uPred_reflection.QuoteArgs _ _ ?ns' => ns'
                end in
@@ -188,7 +188,7 @@ Tactic Notation "to_front" open_constr(Ps) :=
 Tactic Notation "to_back" open_constr(Ps) :=
   close_uPreds Ps ltac:(fun Ps =>
     uPred_reflection.quote_l;
-    let Σ := match goal with |- uPred_reflection.eval ?Σ _ ⊑ _ => Σ end in
+    let Σ := match goal with |- uPred_reflection.eval ?Σ _ ⊢ _ => Σ end in
     let ns' := lazymatch type of (_ : uPred_reflection.QuoteArgs Σ Ps _) with
                | uPred_reflection.QuoteArgs _ _ ?ns' => ns'
                end in
@@ -205,46 +205,46 @@ Tactic Notation "sep_split" "right:" open_constr(Ps) :=
 Tactic Notation "sep_split" "left:" open_constr(Ps) :=
   to_front Ps; apply sep_mono.
 
-(** Assumes a goal of the shape P ⊑ ▷ Q. Alterantively, if Q
+(** Assumes a goal of the shape P ⊢ ▷ Q. Alterantively, if Q
     is built of ★, ∧, ∨ with ▷ in all branches; that will work, too.
-    Will turn this goal into P ⊑ Q and strip ▷ in P below ★, ∧, ∨. *)
+    Will turn this goal into P ⊢ Q and strip ▷ in P below ★, ∧, ∨. *)
 Ltac strip_later :=
   let rec strip :=
     lazymatch goal with
-    | |- (_ ★ _) ⊑ ▷ _  =>
+    | |- (_ ★ _) ⊢ ▷ _  =>
       etrans; last (eapply equiv_entails_sym, later_sep);
       apply sep_mono; strip
-    | |- (_ ∧ _) ⊑ ▷ _  =>
+    | |- (_ ∧ _) ⊢ ▷ _  =>
       etrans; last (eapply equiv_entails_sym, later_and);
       apply sep_mono; strip
-    | |- (_ ∨ _) ⊑ ▷ _  =>
+    | |- (_ ∨ _) ⊢ ▷ _  =>
       etrans; last (eapply equiv_entails_sym, later_or);
       apply sep_mono; strip
-    | |- ▷ _ ⊑ ▷ _ => apply later_mono; reflexivity
-    | |- _ ⊑ ▷ _ => apply later_intro; reflexivity
+    | |- ▷ _ ⊢ ▷ _ => apply later_mono; reflexivity
+    | |- _ ⊢ ▷ _ => apply later_intro; reflexivity
     end
   in let rec shape_Q :=
     lazymatch goal with
-    | |- _ ⊑ (_ ★ _) =>
+    | |- _ ⊢ (_ ★ _) =>
       (* Force the later on the LHS to be top-level, matching laters
          below ★ on the RHS *)
       etrans; first (apply equiv_entails, later_sep; reflexivity);
       (* Match the arm recursively *)
       apply sep_mono; shape_Q
-    | |- _ ⊑ (_ ∧ _) =>
+    | |- _ ⊢ (_ ∧ _) =>
       etrans; first (apply equiv_entails, later_and; reflexivity);
       apply sep_mono; shape_Q
-    | |- _ ⊑ (_ ∨ _) =>
+    | |- _ ⊢ (_ ∨ _) =>
       etrans; first (apply equiv_entails, later_or; reflexivity);
       apply sep_mono; shape_Q
-    | |- _ ⊑ ▷ _ => apply later_mono; reflexivity
+    | |- _ ⊢ ▷ _ => apply later_mono; reflexivity
     (* We fail if we don't find laters in all branches. *)
     end
   in intros_revert ltac:(etrans; [|shape_Q];
                          etrans; last eapply later_mono; first solve [ strip ]).
 
-(** Transforms a goal of the form ∀ ..., ?0... → ?1 ⊑ ?2
-    into True ⊑ ∀..., ■?0... → ?1 → ?2, applies tac, and
+(** Transforms a goal of the form ∀ ..., ?0... → ?1 ⊢ ?2
+    into True ⊢ ∀..., ■?0... → ?1 → ?2, applies tac, and
     the moves all the assumptions back. *)
 (* TODO: this name may be a big too general *)
 Ltac revert_all :=
@@ -256,20 +256,20 @@ Ltac revert_all :=
     on the sort of the argument, which is suboptimal. *)
     first [ apply (const_intro_impl _ _ _ H); clear H
           | revert H; apply forall_elim']
-  | |- _ ⊑ _ => apply impl_entails
+  | |- _ ⊢ _ => apply impl_entails
   end.
 
-(** This starts on a goal of the form ∀ ..., ?0... → ?1 ⊑ ?2.
+(** This starts on a goal of the form ∀ ..., ?0... → ?1 ⊢ ?2.
    It applies löb where all the Coq assumptions have been turned into logical
    assumptions, then moves all the Coq assumptions back out to the context,
-   applies [tac] on the goal (now of the form _ ⊑ _), and then reverts the
+   applies [tac] on the goal (now of the form _ ⊢ _), and then reverts the
    Coq assumption so that we end up with the same shape as where we started,
    but with an additional assumption ★-ed to the context *)
 Ltac löb tac :=
   revert_all;
   (* Add a box *)
   etrans; last (eapply always_elim; reflexivity);
-  (* We now have a goal for the form True ⊑ P, with the "original" conclusion
+  (* We now have a goal for the form True ⊢ P, with the "original" conclusion
      being locked. *)
   apply löb_strong; etransitivity;
     first (apply equiv_entails, left_id, _; reflexivity);
@@ -278,13 +278,13 @@ Ltac löb tac :=
      do the work *)
   let rec go :=
     lazymatch goal with
-    | |- _ ⊑ (∀ _, _) =>
+    | |- _ ⊢ (∀ _, _) =>
       apply forall_intro; let H := fresh in intro H; go; revert H
-    | |- _ ⊑ (■ _ → _) =>
+    | |- _ ⊢ (■ _ → _) =>
       apply impl_intro_l, const_elim_l; let H := fresh in intro H; go; revert H
     (* This is the "bottom" of the goal, where we see the impl introduced
        by uPred_revert_all as well as the ▷ from löb_strong and the □ we added. *)
-    | |- ▷ □ ?R ⊑ (?L → _) => apply impl_intro_l;
+    | |- ▷ □ ?R ⊢ (?L → _) => apply impl_intro_l;
       trans (L ★ ▷ □ R)%I;
         [eapply equiv_entails, always_and_sep_r, _; reflexivity | tac]