Merge uPred_mono and uPred_closed.

theories/base_logic/double_negation.v

@@ -30,11 +30,11 @@ Import uPred.
 Lemma laterN_big n a x φ: ✓{n} x →  a ≤ n → (▷^a ⌜φ⌝)%I n x → φ.
 Proof.
   induction 2 as [| ?? IHle].
-  - induction a; repeat (rewrite //= || uPred.unseal). 
+  - induction a; repeat (rewrite //= || uPred.unseal).
     intros Hlater. apply IHa; auto using cmra_validN_S.
-    move:Hlater; repeat (rewrite //= || uPred.unseal). 
+    move:Hlater; repeat (rewrite //= || uPred.unseal).
   - intros. apply IHle; auto using cmra_validN_S.
-    eapply uPred_closed; eauto using cmra_validN_S.
+    eapply uPred_mono; eauto using cmra_validN_S.
 Qed.
 
 Lemma laterN_small n a x φ: ✓{n} x →  n < a → (▷^a ⌜φ⌝)%I n x.
@@ -46,15 +46,15 @@ Proof.
   - induction n as [| n IHn]; [| move: IHle];
       repeat (rewrite //= || uPred.unseal).
     red; rewrite //=. intros.
-    eapply (uPred_closed _ _ (S n)); eauto using cmra_validN_S.
+    eapply (uPred_mono _ _ (S n)); eauto using cmra_validN_S.
 Qed.
 
 (* It is easy to show that most of the basic properties of bupd that
-   are used throughout Iris hold for nnupd. 
+   are used throughout Iris hold for nnupd.
 
    In fact, the first three properties that follow hold for any
    modality of the form (- -∗ Q) -∗ Q for arbitrary Q. The situation
-   here is slightly different, because nnupd is of the form 
+   here is slightly different, because nnupd is of the form
    ∀ n, (- -∗ (Q n)) -∗ (Q n), but the proofs carry over straightforwardly.
 
  *)
@@ -77,8 +77,8 @@ Proof.
 Qed.
 Lemma nnupd_ownM_updateP x (Φ : M → Prop) :
   x ~~>: Φ → uPred_ownM x =n=> ∃ y, ⌜Φ y⌝ ∧ uPred_ownM y.
-Proof. 
-  intros Hbupd. split. rewrite /uPred_nnupd. repeat uPred.unseal. 
+Proof.
+  intros Hbupd. split. rewrite /uPred_nnupd. repeat uPred.unseal.
   intros n y ? Hown a.
   red; rewrite //= => n' yf ??.
   inversion Hown as (x'&Hequiv).
@@ -87,18 +87,18 @@ Proof.
   case (decide (a ≤ n')).
   - intros Hle Hwand.
     exfalso. eapply laterN_big; last (uPred.unseal; eapply (Hwand n' (y' ⋅ x'))); eauto.
-    * rewrite comm -assoc. done. 
-    * rewrite comm -assoc. done. 
-    * eexists. split; eapply uPred_mono; red; rewrite //=; eauto.
-  - intros; assert (n' < a). omega.
+    * rewrite comm -assoc. done.
+    * rewrite comm -assoc. done.
+    * exists y'. split=>//. by exists x'.
+ - intros; assert (n' < a). omega.
     move: laterN_small. uPred.unseal.
     naive_solver.
 Qed.
 
 (* However, the transitivity property seems to be much harder to
-   prove. This is surprising, because transitivity does hold for 
+   prove. This is surprising, because transitivity does hold for
    modalities of the form (- -∗ Q) -∗ Q. What goes wrong when we quantify
-   now over n? 
+   now over n?
  *)
 
 Remark nnupd_trans P: (|=n=> |=n=> P) ⊢ (|=n=> P).
@@ -111,7 +111,7 @@ Proof.
   (* Oops -- the exponents of the later modality don't match up! *)
 Abort.
 
-(* Instead, we will need to prove this in the model. We start by showing that 
+(* Instead, we will need to prove this in the model. We start by showing that
    nnupd is the limit of a the following sequence:
 
    (- -∗ False) - ∗ False,
@@ -121,12 +121,12 @@ Abort.
 
    Then, it is easy enough to show that each of the uPreds in this sequence
    is transitive. It turns out that this implies that nnupd is transitive. *)
-   
+
 
 (* The definition of the sequence above: *)
 Fixpoint uPred_nnupd_k {M} k (P: uPred M) : uPred M :=
   ((P -∗ ▷^k False) -∗ ▷^k False) ∧
-  match k with 
+  match k with
     O => True
   | S k' => uPred_nnupd_k k' P
   end.
@@ -138,11 +138,11 @@ Notation "|=n=>_ k Q" := (uPred_nnupd_k k Q)
 (* One direction of the limiting process is easy -- nnupd implies nnupd_k for each k *)
 Lemma nnupd_trunc1 k P: (|=n=> P) ⊢ |=n=>_k P.
 Proof.
-  induction k. 
-  - rewrite /uPred_nnupd_k /uPred_nnupd. 
+  induction k.
+  - rewrite /uPred_nnupd_k /uPred_nnupd.
     rewrite (forall_elim 0) //= right_id //.
   - simpl. apply and_intro; auto.
-    rewrite /uPred_nnupd. 
+    rewrite /uPred_nnupd.
     rewrite (forall_elim (S k)) //=.
 Qed.
 
@@ -190,10 +190,10 @@ Lemma nnupd_nnupd_k_dist k P: (|=n=> P)%I ≡{k}≡ (|=n=>_k P)%I.
          *** intros. exfalso. assert (n ≤ k'). omega.
              assert (n = S k ∨ n < S k) as [->|] by omega.
              **** eapply laterN_big; eauto; unseal. eapply HnnP; eauto.
-             **** move:nnupd_k_elim. unseal. intros Hnnupdk. 
+             **** move:nnupd_k_elim. unseal. intros Hnnupdk.
                   eapply laterN_big; eauto. unseal.
                   eapply (Hnnupdk n k); first omega; eauto.
-                  exists x, x'. split_and!; eauto. eapply uPred_closed; eauto.
+                  exists x, x'. split_and!; eauto. eapply uPred_mono; eauto.
       ** intros HP. eapply IHk; auto.
          move:HP. unseal. intros (?&?); naive_solver.
 Qed.
@@ -203,13 +203,13 @@ Lemma nnupd_k_intro k P: P ⊢ (|=n=>_k P).
 Proof.
   induction k; rewrite //= ?right_id.
   - apply wand_intro_l. apply wand_elim_l.
-  - apply and_intro; auto. 
+  - apply and_intro; auto.
     apply wand_intro_l. apply wand_elim_l.
 Qed.
 
 Lemma nnupd_k_mono k P Q: (P ⊢ Q) → (|=n=>_k P) ⊢ (|=n=>_k Q).
 Proof.
-  induction k; rewrite //= ?right_id=>HPQ. 
+  induction k; rewrite //= ?right_id=>HPQ.
   - do 2 (apply wand_mono; auto).
   - apply and_mono; auto; do 2 (apply wand_mono; auto).
 Qed.
@@ -227,7 +227,7 @@ Lemma nnupd_k_trans k P: (|=n=>_k |=n=>_k P) ⊢ (|=n=>_k P).
 Proof.
   revert P.
   induction k; intros P.
-  - rewrite //= ?right_id. apply wand_intro_l. 
+  - rewrite //= ?right_id. apply wand_intro_l.
     rewrite {1}(nnupd_k_intro 0 (P -∗ False)%I) //= ?right_id. apply wand_elim_r. 
   - rewrite {2}(nnupd_k_unfold k P).
     apply and_intro.
@@ -262,8 +262,8 @@ Proof.
   case (decide (a ≤ n')).
   - intros Hle Hwand.
     exfalso. eapply laterN_big; last (uPred.unseal; eapply (Hwand n' x')); eauto.
-    * rewrite comm. done. 
-    * rewrite comm. done. 
+    * rewrite comm. done.
+    * rewrite comm. done.
   - intros; assert (n' < a). omega.
     move: laterN_small. uPred.unseal.
     naive_solver.
@@ -299,23 +299,23 @@ End classical.
 Lemma nnupd_dne φ: (|=n=> ⌜¬¬ φ → φ⌝: uPred M)%I.
 Proof.
   rewrite /uPred_nnupd. apply forall_intro=>n.
-  apply wand_intro_l. rewrite ?right_id. 
+  apply wand_intro_l. rewrite ?right_id.
   assert (∀ φ, ¬¬¬¬φ → ¬¬φ) by naive_solver.
   assert (Hdne: ¬¬ (¬¬φ → φ)) by naive_solver.
   split. unseal. intros n' ?? Hupd.
   case (decide (n' < n)).
   - intros. move: laterN_small. unseal. naive_solver.
-  - intros. assert (n ≤ n'). omega. 
+  - intros. assert (n ≤ n'). omega.
     exfalso. specialize (Hupd n' ε).
     eapply Hdne. intros Hfal.
-    eapply laterN_big; eauto. 
+    eapply laterN_big; eauto.
     unseal. rewrite right_id in Hupd *; naive_solver.
 Qed.
 
 (* Nevertheless, we can prove a weaker form of adequacy (which is equvialent to adequacy
    under classical axioms) directly without passing through the proofs for bupd: *)
 Lemma adequacy_helper1 P n k x:
-  ✓{S n + k} x → ¬¬ (Nat.iter (S n) (λ P, |=n=> ▷ P)%I P (S n + k) x) 
+  ✓{S n + k} x → ¬¬ (Nat.iter (S n) (λ P, |=n=> ▷ P)%I P (S n + k) x)
             → ¬¬ (∃ x', ✓{n + k} (x') ∧ Nat.iter n (λ P, |=n=> ▷ P)%I P (n + k) (x')).
 Proof.
   revert k P x. induction n.
@@ -325,12 +325,12 @@ Proof.
     specialize (Hf3 (S k) (S k) ε). rewrite right_id in Hf3 *. unseal.
     intros Hf3. eapply Hf3; eauto.
     intros ??? Hx'. rewrite left_id in Hx' *=> Hx'.
-    unseal. 
+    unseal.
     assert (n' < S k ∨ n' = S k) as [|] by omega.
     * intros. move:(laterN_small n' (S k) x' False). rewrite //=. unseal. intros Hsmall. 
       eapply Hsmall; eauto.
     * subst. intros. exfalso. eapply Hf2. exists x'. split; eauto using cmra_validN_S.
-  - intros k P x Hx. rewrite ?Nat_iter_S_r. 
+  - intros k P x Hx. rewrite ?Nat_iter_S_r.
     replace (S (S n) + k) with (S n + (S k)) by omega.
     replace (S n + k) with (n + (S k)) by omega.
     intros. eapply IHn. replace (S n + S k) with (S (S n) + k) by omega. eauto.
@@ -338,7 +338,7 @@ Proof.
 Qed.
 
 Lemma adequacy_helper2 P n k x:
-  ✓{S n + k} x → ¬¬ (Nat.iter (S n) (λ P, |=n=> ▷ P)%I P (S n + k) x) 
+  ✓{S n + k} x → ¬¬ (Nat.iter (S n) (λ P, |=n=> ▷ P)%I P (S n + k) x)
             → ¬¬ (∃ x', ✓{k} (x') ∧ Nat.iter 0 (λ P, |=n=> ▷ P)%I P k (x')).
 Proof.
   revert x. induction n.

theories/base_logic/primitive.v

@@ -35,11 +35,10 @@ Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∀ n' x',
        x ≼ x' → n' ≤ n → ✓{n'} x' → P n' x' → Q n' x' |}.
 Next Obligation.
-  intros M P Q n1 x1 x1' HPQ [x2 Hx1'] n2 x3 [x4 Hx3] ?; simpl in *.
+  intros M P Q n1 n1' x1 x1' HPQ [x2 Hx1'] Hn1 n2 x3 [x4 Hx3] ?; simpl in *.
   rewrite Hx3 (dist_le _ _ _ _ Hx1'); auto. intros ??.
   eapply HPQ; auto. exists (x2 ⋅ x4); by rewrite assoc.
 Qed.
-Next Obligation. intros M P Q [|n1] [|n2] x; auto with lia. Qed.
 Definition uPred_impl_aux : seal (@uPred_impl_def). by eexists. Qed.
 Definition uPred_impl {M} := unseal uPred_impl_aux M.
 Definition uPred_impl_eq :
@@ -71,14 +70,9 @@ Definition uPred_internal_eq_eq:
 Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}.
 Next Obligation.
-  intros M P Q n x y (x1&x2&Hx&?&?) [z Hy].
+  intros M P Q n1 n2 x y (x1&x2&Hx&?&?) [z Hy] Hn.
   exists x1, (x2 ⋅ z); split_and?; eauto using uPred_mono, cmra_includedN_l.
-  by rewrite Hy Hx assoc.
-Qed.
-Next Obligation.
-  intros M P Q n1 n2 x (x1&x2&Hx&?&?) ?.
-  exists x1, x2; ofe_subst; split_and!;
-    eauto using dist_le, uPred_closed, cmra_validN_op_l, cmra_validN_op_r.
+  eapply dist_le, Hn. by rewrite Hy Hx assoc.
 Qed.
 Definition uPred_sep_aux : seal (@uPred_sep_def). by eexists. Qed.
 Definition uPred_sep {M} := unseal uPred_sep_aux M.
@@ -88,11 +82,10 @@ Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∀ n' x',
        n' ≤ n → ✓{n'} (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}.
 Next Obligation.
-  intros M P Q n x1 x1' HPQ ? n3 x3 ???; simpl in *.
-  apply uPred_mono with (x1 ⋅ x3);
+  intros M P Q n1 n1' x1 x1' HPQ ? Hn n3 x3 ???; simpl in *.
+  eapply uPred_mono with n3 (x1 ⋅ x3);
     eauto using cmra_validN_includedN, cmra_monoN_r, cmra_includedN_le.
 Qed.
-Next Obligation. naive_solver. Qed.
 Definition uPred_wand_aux : seal (@uPred_wand_def). by eexists. Qed.
 Definition uPred_wand {M} := unseal uPred_wand_aux M.
 Definition uPred_wand_eq :
@@ -103,7 +96,7 @@ Definition uPred_wand_eq :
    because Iris is afine.  The following is easier to work with. *)
 Program Definition uPred_plainly_def {M} (P : uPred M) : uPred M :=
   {| uPred_holds n x := P n ε |}.
-Solve Obligations with naive_solver eauto using uPred_closed, ucmra_unit_validN.
+Solve Obligations with naive_solver eauto using uPred_mono, ucmra_unit_validN.
 Definition uPred_plainly_aux : seal (@uPred_plainly_def). by eexists. Qed.
 Definition uPred_plainly {M} := unseal uPred_plainly_aux M.
 Definition uPred_plainly_eq :
@@ -114,7 +107,6 @@ Program Definition uPred_persistently_def {M} (P : uPred M) : uPred M :=
 Next Obligation.
   intros M; naive_solver eauto using uPred_mono, @cmra_core_monoN.
 Qed.
-Next Obligation. naive_solver eauto using uPred_closed, @cmra_core_validN. Qed.
 Definition uPred_persistently_aux : seal (@uPred_persistently_def). by eexists. Qed.
 Definition uPred_persistently {M} := unseal uPred_persistently_aux M.
 Definition uPred_persistently_eq :
@@ -123,10 +115,7 @@ Definition uPred_persistently_eq :
 Program Definition uPred_later_def {M} (P : uPred M) : uPred M :=
   {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
 Next Obligation.
-  intros M P [|n] x1 x2; eauto using uPred_mono, cmra_includedN_S.
-Qed.
-Next Obligation.
-  intros M P [|n1] [|n2] x; eauto using uPred_closed, cmra_validN_S with lia.
+  intros M P [|n1] [|n2] x1 x2; eauto using uPred_mono, cmra_includedN_S with lia.
 Qed.
 Definition uPred_later_aux : seal (@uPred_later_def). by eexists. Qed.
 Definition uPred_later {M} := unseal uPred_later_aux M.
@@ -136,10 +125,9 @@ Definition uPred_later_eq :
 Program Definition uPred_ownM_def {M : ucmraT} (a : M) : uPred M :=
   {| uPred_holds n x := a ≼{n} x |}.
 Next Obligation.
-  intros M a n x1 x [a' Hx1] [x2 ->].
-  exists (a' ⋅ x2). by rewrite (assoc op) Hx1.
+  intros M a n1 n2 x1 x [a' Hx1] [x2 Hx] Hn. eapply cmra_includedN_le=>//.
+  exists (a' ⋅ x2). by rewrite Hx(assoc op) Hx1.
 Qed.
-Next Obligation. naive_solver eauto using cmra_includedN_le. Qed.
 Definition uPred_ownM_aux : seal (@uPred_ownM_def). by eexists. Qed.
 Definition uPred_ownM {M} := unseal uPred_ownM_aux M.
 Definition uPred_ownM_eq :
@@ -157,13 +145,12 @@ Program Definition uPred_bupd_def {M} (Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∀ k yf,
       k ≤ n → ✓{k} (x ⋅ yf) → ∃ x', ✓{k} (x' ⋅ yf) ∧ Q k x' |}.
 Next Obligation.
-  intros M Q n x1 x2 HQ [x3 Hx] k yf Hk.
+  intros M Q n1 n2 x1 x2 HQ [x3 Hx] Hn k yf Hk.
   rewrite (dist_le _ _ _ _ Hx); last lia. intros Hxy.
   destruct (HQ k (x3 ⋅ yf)) as (x'&?&?); [auto|by rewrite assoc|].
   exists (x' ⋅ x3); split; first by rewrite -assoc.
-  apply uPred_mono with x'; eauto using cmra_includedN_l.
+  eauto using uPred_mono, cmra_includedN_l.
 Qed.
-Next Obligation. naive_solver. Qed.
 Definition uPred_bupd_aux : seal (@uPred_bupd_def). by eexists. Qed.
 Definition uPred_bupd {M} := unseal uPred_bupd_aux M.
 Definition uPred_bupd_eq : @uPred_bupd = @uPred_bupd_def := seal_eq uPred_bupd_aux.
@@ -380,7 +367,7 @@ Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed.
 Lemma impl_intro_r P Q R : (P ∧ Q ⊢ R) → P ⊢ Q → R.
 Proof.
   unseal; intros HQ; split=> n x ?? n' x' ????. apply HQ;
-    naive_solver eauto using uPred_mono, uPred_closed, cmra_included_includedN.
+    naive_solver eauto using uPred_mono, cmra_included_includedN.
 Qed.
 Lemma impl_elim P Q R : (P ⊢ Q → R) → (P ⊢ Q) → P ⊢ R.
 Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed.
@@ -432,7 +419,7 @@ Lemma wand_intro_r P Q R : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R.
 Proof.
   unseal=> HPQR; split=> n x ?? n' x' ???; apply HPQR; auto.
   exists x, x'; split_and?; auto.
-  eapply uPred_closed with n; eauto using cmra_validN_op_l.
+  eapply uPred_mono with n x; eauto using cmra_validN_op_l.
 Qed.
 Lemma wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R.
 Proof.
@@ -486,14 +473,14 @@ Qed.
 Lemma persistently_impl_plainly P Q : (■ P → □ Q) ⊢ □ (■ P → Q).
 Proof.
   unseal; split=> /= n x ? HPQ n' x' ????.
-  eapply uPred_mono with (core x), cmra_included_includedN; auto.
+  eapply uPred_mono with n' (core x)=>//; [|by apply cmra_included_includedN].
   apply (HPQ n' x); eauto using cmra_validN_le.
 Qed.
 
 Lemma plainly_impl_plainly P Q : (■ P → ■ Q) ⊢ ■ (■ P → Q).
 Proof.
   unseal; split=> /= n x ? HPQ n' x' ????.
-  eapply uPred_mono with ε, cmra_included_includedN; auto.
+  eapply uPred_mono with n' ε=>//; [|by apply cmra_included_includedN].
   apply (HPQ n' x); eauto using cmra_validN_le.
 Qed.
 
@@ -505,7 +492,7 @@ Qed.
 Lemma löb P : (▷ P → P) ⊢ P.
 Proof.
   unseal; split=> n x ? HP; induction n as [|n IH]; [by apply HP|].
-  apply HP, IH, uPred_closed with (S n); eauto using cmra_validN_S.
+  apply HP, IH, uPred_mono with (S n) x; eauto using cmra_validN_S.
 Qed.
 Lemma later_forall_2 {A} (Φ : A → uPred M) : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a.
 Proof. unseal; by split=> -[|n] x. Qed.
@@ -526,8 +513,7 @@ Qed.
 Lemma later_false_excluded_middle P : ▷ P ⊢ ▷ False ∨ (▷ False → P).
 Proof.
   unseal; split=> -[|n] x ? /= HP; [by left|right].
-  intros [|n'] x' ????; [|done].
-  eauto using uPred_closed, uPred_mono, cmra_included_includedN.
+  intros [|n'] x' ????; eauto using uPred_mono, cmra_included_includedN.
 Qed.
 Lemma persistently_later P : □ ▷ P ⊣⊢ ▷ □ P.
 Proof. by unseal. Qed.
@@ -577,7 +563,7 @@ Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed.
 Lemma bupd_intro P : P ==∗ P.
 Proof.
   unseal. split=> n x ? HP k yf ?; exists x; split; first done.
-  apply uPred_closed with n; eauto using cmra_validN_op_l.
+  apply uPred_mono with n x; eauto using cmra_validN_op_l.
 Qed.
 Lemma bupd_mono P Q : (P ⊢ Q) → (|==> P) ==∗ Q.
 Proof.
@@ -593,8 +579,7 @@ Proof.
   destruct (HP k (x2 ⋅ yf)) as (x'&?&?); eauto.
   { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. }
   exists (x' ⋅ x2); split; first by rewrite -assoc.
-  exists x', x2; split_and?; auto.
-  apply uPred_closed with n; eauto 3 using cmra_validN_op_l, cmra_validN_op_r.
+  exists x', x2. eauto using uPred_mono, cmra_validN_op_l, cmra_validN_op_r.
 Qed.
 Lemma bupd_ownM_updateP x (Φ : M → Prop) :
   x ~~>: Φ → uPred_ownM x ==∗ ∃ y, ⌜Φ y⌝ ∧ uPred_ownM y.

theories/base_logic/upred.v

@@ -9,9 +9,8 @@ Set Default Proof Using "Type".
 Record uPred (M : ucmraT) : Type := IProp {
   uPred_holds :> nat → M → Prop;
 
-  uPred_mono n x1 x2 : uPred_holds n x1 → x1 ≼{n} x2 → uPred_holds n x2;
-
-  uPred_closed n1 n2 x : uPred_holds n1 x → n2 ≤ n1 → uPred_holds n2 x
+  uPred_mono n1 n2 x1 x2 :
+    uPred_holds n1 x1 → x1 ≼{n1} x2 → n2 ≤ n1 → uPred_holds n2 x2
 }.
 Arguments uPred_holds {_} _ _ _ : simpl never.
 Add Printing Constructor uPred.
@@ -81,18 +80,15 @@ Section cofe.
   Program Definition uPred_compl : Compl uPredC := λ c,
     {| uPred_holds n x := ∀ n', n' ≤ n → ✓{n'}x → c n' n' x |}.
   Next Obligation.
-    move=> /= c n x1 x2 Hx1 Hx12 n' Hn'n Hv.
-    eapply uPred_mono. by eapply Hx1, cmra_validN_includedN, cmra_includedN_le.
-    by eapply cmra_includedN_le.
-  Qed.
-  Next Obligation.
-    move=> /= c n1 n2 x Hc Hn12 n3 Hn23 Hv. apply Hc. lia. done.
+    move=> /= c n1 n2 x1 x2 HP Hx12 Hn12 n3 Hn23 Hv. eapply uPred_mono.
+    eapply HP, cmra_validN_includedN, cmra_includedN_le=>//; lia.
+    eapply cmra_includedN_le=>//; lia. done.
   Qed.
   Global Program Instance uPred_cofe : Cofe uPredC := {| compl := uPred_compl |}.
   Next Obligation.
     intros n c; split=>i x Hin Hv.
     etrans; [|by symmetry; apply (chain_cauchy c i n)]. split=>H; [by apply H|].
-    repeat intro. apply (chain_cauchy c n' i)=>//. by eapply uPred_closed.
+    repeat intro. apply (chain_cauchy c n' i)=>//. by eapply uPred_mono.
   Qed.
 End cofe.
 Arguments uPredC : clear implicits.
@@ -107,8 +103,7 @@ Proof. by intros x1 x2 Hx; apply uPred_ne, equiv_dist. Qed.
 Lemma uPred_holds_ne {M} (P Q : uPred M) n1 n2 x :
   P ≡{n2}≡ Q → n2 ≤ n1 → ✓{n2} x → Q n1 x → P n2 x.
 Proof.
-  intros [Hne] ???. eapply Hne; try done.
-  eapply uPred_closed; eauto using cmra_validN_le.
+  intros [Hne] ???. eapply Hne; try done. eauto using uPred_mono, cmra_validN_le.
 Qed.
 
 (** functor *)
@@ -116,7 +111,6 @@ Program Definition uPred_map {M1 M2 : ucmraT} (f : M2 -n> M1)
   `{!CmraMorphism f} (P : uPred M1) :
   uPred M2 := {| uPred_holds n x := P n (f x) |}.
 Next Obligation. naive_solver eauto using uPred_mono, cmra_morphism_monotoneN. Qed.
-Next Obligation. naive_solver eauto using uPred_closed, cmra_morphism_validN. Qed.
 
 Instance uPred_map_ne {M1 M2 : ucmraT} (f : M2 -n> M1)
   `{!CmraMorphism f} n : Proper (dist n ==> dist n) (uPred_map f).
@@ -174,7 +168,7 @@ Inductive uPred_entails {M} (P Q : uPred M) : Prop :=
 Hint Extern 0 (uPred_entails _ _) => reflexivity.
 Instance uPred_entails_rewrite_relation M : RewriteRelation (@uPred_entails M).
 
-Hint Resolve uPred_mono uPred_closed : uPred_def.
+Hint Resolve uPred_mono : uPred_def.
 
 (** Notations *)
 Notation "P ⊢ Q" := (uPred_entails P%I Q%I)