Rename iProp into uPred.

45aa85cb · Robbert Krebbers · 0f025d83 · 45aa85cb
Commit 45aa85cb authored Nov 16, 2015 by Robbert Krebbers
--- a/iris/logic.v
+++ b/iris/logic.v
@@ -3,27 +3,27 @@ Local Hint Extern 1 (_ ≼ _) => etransitivity; [eassumption|].
 Local Hint Extern 1 (_ ≼ _) => etransitivity; [|eassumption].
 Local Hint Extern 10 (_ ≤ _) => omega.

-Structure iProp (M : cmraT) : Type := IProp {
-  iprop_holds :> nat → M -> Prop;
-  iprop_ne x1 x2 n : iprop_holds n x1 → x1 ={n}= x2 → iprop_holds n x2;
-  iprop_weaken x1 x2 n1 n2 :
-    x1 ≼ x2 → n2 ≤ n1 → validN n2 x2 → iprop_holds n1 x1 → iprop_holds n2 x2
+Structure uPred (M : cmraT) : Type := IProp {
+  uPred_holds :> nat → M -> Prop;
+  uPred_ne x1 x2 n : uPred_holds n x1 → x1 ={n}= x2 → uPred_holds n x2;
+  uPred_weaken x1 x2 n1 n2 :
+    x1 ≼ x2 → n2 ≤ n1 → validN n2 x2 → uPred_holds n1 x1 → uPred_holds n2 x2
 }.
-Add Printing Constructor iProp.
-Instance: Params (@iprop_holds) 3.
+Add Printing Constructor uPred.
+Instance: Params (@uPred_holds) 3.

-Instance iprop_equiv (M : cmraT) : Equiv (iProp M) := λ P Q, ∀ x n,
+Instance uPred_equiv (M : cmraT) : Equiv (uPred M) := λ P Q, ∀ x n,
  validN n x → P n x ↔ Q n x.
-Instance iprop_dist (M : cmraT) : Dist (iProp M) := λ n P Q, ∀ x n',
+Instance uPred_dist (M : cmraT) : Dist (uPred M) := λ n P Q, ∀ x n',
  n' < n → validN n' x → P n' x ↔ Q n' x.
-Program Instance iprop_compl (M : cmraT) : Compl (iProp M) := λ c,
-  {| iprop_holds n x := c (S n) n x |}.
-Next Obligation. by intros M c x y n ??; simpl in *; apply iprop_ne with x. Qed.
+Program Instance uPred_compl (M : cmraT) : Compl (uPred M) := λ c,
+  {| uPred_holds n x := c (S n) n x |}.
+Next Obligation. by intros M c x y n ??; simpl in *; apply uPred_ne with x. Qed.
 Next Obligation.
  intros M c x1 x2 n1 n2 ????; simpl in *.
-  apply (chain_cauchy c (S n2) (S n1)); eauto using iprop_weaken, cmra_valid_le.
+  apply (chain_cauchy c (S n2) (S n1)); eauto using uPred_weaken, cmra_valid_le.
 Qed.
-Instance iprop_cofe (M : cmraT) : Cofe (iProp M).
+Instance uPred_cofe (M : cmraT) : Cofe (uPred M).
 Proof.
  split.
  * intros P Q; split; [by intros HPQ n x i ??; apply HPQ|].
@@ -36,70 +36,70 @@ Proof.
  * intros P Q x i ??; lia.
  * intros c n x i ??; apply (chain_cauchy c (S i) n); auto.
 Qed.
-Instance iprop_holds_ne {M} (P : iProp M) n : Proper (dist n ==> iff) (P n).
-Proof. intros x1 x2 Hx; split; eauto using iprop_ne. Qed.
-Instance iprop_holds_proper {M} (P : iProp M) n : Proper ((≡) ==> iff) (P n).
-Proof. by intros x1 x2 Hx; apply iprop_holds_ne, equiv_dist. Qed.
-Definition iPropC (M : cmraT) : cofeT := CofeT (iProp M).
+Instance uPred_holds_ne {M} (P : uPred M) n : Proper (dist n ==> iff) (P n).
+Proof. intros x1 x2 Hx; split; eauto using uPred_ne. Qed.
+Instance uPred_holds_proper {M} (P : uPred M) n : Proper ((≡) ==> iff) (P n).
+Proof. by intros x1 x2 Hx; apply uPred_holds_ne, equiv_dist. Qed.
+Definition uPredC (M : cmraT) : cofeT := CofeT (uPred M).

 (** functor *)
-Program Definition iprop_map {M1 M2 : cmraT} (f : M2 → M1)
+Program Definition uPred_map {M1 M2 : cmraT} (f : M2 → M1)
  `{!∀ n, Proper (dist n ==> dist n) f, !CMRAPreserving f}
-  (P : iProp M1) : iProp M2 := {| iprop_holds n x := P n (f x) |}.
+  (P : uPred M1) : uPred M2 := {| uPred_holds n x := P n (f x) |}.
 Next Obligation. by intros M1 M2 f ?? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed.
 Next Obligation.
-  by intros M1 M2 f ?? P y1 y2 n i ???; simpl; apply iprop_weaken; auto;
+  by intros M1 M2 f ?? P y1 y2 n i ???; simpl; apply uPred_weaken; auto;
    apply validN_preserving || apply included_preserving.
 Qed.
-Instance iprop_map_ne {M1 M2 : cmraT} (f : M2 → M1)
+Instance uPred_map_ne {M1 M2 : cmraT} (f : M2 → M1)
  `{!∀ n, Proper (dist n ==> dist n) f, !CMRAPreserving f} :
-  Proper (dist n ==> dist n) (iprop_map f).
+  Proper (dist n ==> dist n) (uPred_map f).
 Proof.
  by intros n x1 x2 Hx y n'; split; apply Hx; try apply validN_preserving.
 Qed.
-Definition ipropC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAPreserving f} :
-  iPropC M1 -n> iPropC M2 := CofeMor (iprop_map f : iPropC M1 → iPropC M2).
+Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAPreserving f} :
+  uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2).

 (** logical entailement *)
-Instance iprop_entails {M} : SubsetEq (iProp M) := λ P Q, ∀ x n,
+Instance uPred_entails {M} : SubsetEq (uPred M) := λ P Q, ∀ x n,
  validN n x → P n x → Q n x.

 (** logical connectives *)
-Program Definition iprop_const {M} (P : Prop) : iProp M :=
-  {| iprop_holds n x := P |}.
+Program Definition uPred_const {M} (P : Prop) : uPred M :=
+  {| uPred_holds n x := P |}.
 Solve Obligations with done.

-Program Definition iprop_and {M} (P Q : iProp M) : iProp M :=
-  {| iprop_holds n x := P n x ∧ Q n x |}.
-Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
-Program Definition iprop_or {M} (P Q : iProp M) : iProp M :=
-  {| iprop_holds n x := P n x ∨ Q n x |}.
-Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
-Program Definition iprop_impl {M} (P Q : iProp M) : iProp M :=
-  {| iprop_holds n x := ∀ x' n',
+Program Definition uPred_and {M} (P Q : uPred M) : uPred M :=
+  {| uPred_holds n x := P n x ∧ Q n x |}.
+Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
+Program Definition uPred_or {M} (P Q : uPred M) : uPred M :=
+  {| uPred_holds n x := P n x ∨ Q n x |}.
+Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
+Program Definition uPred_impl {M} (P Q : uPred M) : uPred M :=
+  {| uPred_holds n x := ∀ x' n',
       x ≼ x' → n' ≤ n → validN n' x' → P n' x' → Q n' x' |}.
 Next Obligation.
  intros M P Q x1' x1 n1 HPQ Hx1 x2 n2 ????.
  destruct (cmra_included_dist_l x1 x2 x1' n1) as (x2'&?&Hx2); auto.
  assert (x2' ={n2}= x2) as Hx2' by (by apply dist_le with n1).
  assert (validN n2 x2') by (by rewrite Hx2'); rewrite <-Hx2'.
-  by apply HPQ, iprop_weaken with x2' n2, iprop_ne with x2.
+  by apply HPQ, uPred_weaken with x2' n2, uPred_ne with x2.
 Qed.
 Next Obligation. naive_solver eauto 2 with lia. Qed.

-Program Definition iprop_forall {M A} (P : A → iProp M) : iProp M :=
-  {| iprop_holds n x := ∀ a, P a n x |}.
-Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
-Program Definition iprop_exist {M A} (P : A → iProp M) : iProp M :=
-  {| iprop_holds n x := ∃ a, P a n x |}.
-Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
+Program Definition uPred_forall {M A} (P : A → uPred M) : uPred M :=
+  {| uPred_holds n x := ∀ a, P a n x |}.
+Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
+Program Definition uPred_exist {M A} (P : A → uPred M) : uPred M :=
+  {| uPred_holds n x := ∃ a, P a n x |}.
+Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.

-Program Definition iprop_eq {M} {A : cofeT} (a1 a2 : A) : iProp M :=
-  {| iprop_holds n x := a1 ={n}= a2 |}.
+Program Definition uPred_eq {M} {A : cofeT} (a1 a2 : A) : uPred M :=
+  {| uPred_holds n x := a1 ={n}= a2 |}.
 Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).

-Program Definition iprop_sep {M} (P Q : iProp M) : iProp M :=
-  {| iprop_holds n x := ∃ x1 x2, x ={n}= x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}.
+Program Definition uPred_sep {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.
  by intros M P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy.
 Qed.
@@ -110,14 +110,14 @@ Next Obligation.
    exists (x2 ⋅ z); split; eauto using ra_included_l.
    apply dist_le with n1; auto. by rewrite (associative op), <-Hx, Hy. }
  exists x1, x2'; split_ands; auto.
-  * apply iprop_weaken with x1 n1; auto.
+  * apply uPred_weaken with x1 n1; auto.
    by apply cmra_valid_op_l with x2'; rewrite <-Hy.
-  * apply iprop_weaken with x2 n1; auto.
+  * apply uPred_weaken with x2 n1; auto.
    by apply cmra_valid_op_r with x1; rewrite <-Hy.
 Qed.

-Program Definition iprop_wand {M} (P Q : iProp M) : iProp M :=
-  {| iprop_holds n x := ∀ x' n',
+Program Definition uPred_wand {M} (P Q : uPred M) : uPred M :=
+  {| uPred_holds n x := ∀ x' n',
       n' ≤ n → validN n' (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}.
 Next Obligation.
  intros M P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *.
@@ -126,65 +126,65 @@ Next Obligation.
 Qed.
 Next Obligation.
  intros M P Q x1 x2 n1 n2 ??? HPQ x3 n3 ???; simpl in *.
-  apply iprop_weaken with (x1 ⋅ x3) n3; auto using ra_preserving_r.
+  apply uPred_weaken with (x1 ⋅ x3) n3; auto using ra_preserving_r.
  apply HPQ; auto.
  apply cmra_valid_included with (x2 ⋅ x3); auto using ra_preserving_r.
 Qed.

-Program Definition iprop_later {M} (P : iProp M) : iProp M :=
-  {| iprop_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
-Next Obligation. intros M P ?? [|n]; eauto using iprop_ne,(dist_le (A:=M)). Qed.
+Program Definition uPred_later {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]; eauto using uPred_ne,(dist_le (A:=M)). Qed.
 Next Obligation.
  intros M P x1 x2 [|n1] [|n2] ????; auto with lia.
-  apply iprop_weaken with x1 n1; eauto using cmra_valid_S.
+  apply uPred_weaken with x1 n1; eauto using cmra_valid_S.
 Qed.
-Program Definition iprop_always {M} (P : iProp M) : iProp M :=
-  {| iprop_holds n x := P n (unit x) |}.
+Program Definition uPred_always {M} (P : uPred M) : uPred M :=
+  {| uPred_holds n x := P n (unit x) |}.
 Next Obligation. by intros M P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed.
 Next Obligation.
-  intros M P x1 x2 n1 n2 ????; eapply iprop_weaken with (unit x1) n1;
+  intros M P x1 x2 n1 n2 ????; eapply uPred_weaken with (unit x1) n1;
    auto using ra_unit_preserving, cmra_unit_valid.
 Qed.

-Program Definition iprop_own {M : cmraT} (a : M) : iProp M :=
-  {| iprop_holds n x := ∃ a', x ={n}= a ⋅ a' |}.
+Program Definition uPred_own {M : cmraT} (a : M) : uPred M :=
+  {| uPred_holds n x := ∃ a', x ={n}= a ⋅ a' |}.
 Next Obligation. by intros M a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed.
 Next Obligation.
  intros M a x1 x n1 n2; rewrite ra_included_spec; intros [x2 Hx] ?? [a' Hx1].
  exists (a' ⋅ x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx.
 Qed.
-Program Definition iprop_valid {M : cmraT} (a : M) : iProp M :=
-  {| iprop_holds n x := validN n a |}.
+Program Definition uPred_valid {M : cmraT} (a : M) : uPred M :=
+  {| uPred_holds n x := validN n a |}.
 Solve Obligations with naive_solver eauto 2 using cmra_valid_le.

-Definition iprop_fixpoint {M} (P : iProp M → iProp M)
-  `{!Contractive P} : iProp M := fixpoint P (iprop_const True).
+Definition uPred_fixpoint {M} (P : uPred M → uPred M)
+  `{!Contractive P} : uPred M := fixpoint P (uPred_const True).

-Delimit Scope iprop_scope with I.
-Bind Scope iprop_scope with iProp.
-Arguments iprop_holds {_} _%I _ _.
+Delimit Scope uPred_scope with I.
+Bind Scope uPred_scope with uPred.
+Arguments uPred_holds {_} _%I _ _.

-Notation "'False'" := (iprop_const False) : iprop_scope.
-Notation "'True'" := (iprop_const True) : iprop_scope.
-Infix "∧" := iprop_and : iprop_scope.
-Infix "∨" := iprop_or : iprop_scope.
-Infix "→" := iprop_impl : iprop_scope.
-Infix "★" := iprop_sep (at level 80, right associativity) : iprop_scope.
-Infix "-★" := iprop_wand (at level 90) : iprop_scope.
+Notation "'False'" := (uPred_const False) : uPred_scope.
+Notation "'True'" := (uPred_const True) : uPred_scope.
+Infix "∧" := uPred_and : uPred_scope.
+Infix "∨" := uPred_or : uPred_scope.
+Infix "→" := uPred_impl : uPred_scope.
+Infix "★" := uPred_sep (at level 80, right associativity) : uPred_scope.
+Infix "-★" := uPred_wand (at level 90) : uPred_scope.
 Notation "∀ x .. y , P" :=
-  (iprop_forall (λ x, .. (iprop_forall (λ y, P)) ..)) : iprop_scope.
+  (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)) : uPred_scope.
 Notation "∃ x .. y , P" :=
-  (iprop_exist (λ x, .. (iprop_exist (λ y, P)) ..)) : iprop_scope.
-Notation "▷ P" := (iprop_later P) (at level 20) : iprop_scope.
-Notation "□ P" := (iprop_always P) (at level 20) : iprop_scope.
+  (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)) : uPred_scope.
+Notation "▷ P" := (uPred_later P) (at level 20) : uPred_scope.
+Notation "□ P" := (uPred_always P) (at level 20) : uPred_scope.

 Section logic.
 Context {M : cmraT}.
-Implicit Types P Q : iProp M.
+Implicit Types P Q : uPred M.

-Global Instance iprop_preorder : PreOrder ((⊆) : relation (iProp M)).
+Global Instance uPred_preorder : PreOrder ((⊆) : relation (uPred M)).
 Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
-Lemma iprop_equiv_spec P Q : P ≡ Q ↔ P ⊆ Q ∧ Q ⊆ P.
+Lemma uPred_equiv_spec P Q : P ≡ Q ↔ P ⊆ Q ∧ Q ⊆ P.
 Proof.
  split.
  * intros HPQ; split; intros x i; apply HPQ.
@@ -192,142 +192,142 @@ Proof.
 Qed.

 (** Non-expansiveness *)
-Global Instance iprop_const_proper : Proper (iff ==> (≡)) (@iprop_const M).
+Global Instance uPred_const_proper : Proper (iff ==> (≡)) (@uPred_const M).
 Proof. intros P Q HPQ ???; apply HPQ. Qed.
-Global Instance iprop_and_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@iprop_and M).
+Global Instance uPred_and_ne n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
 Proof.
  intros P P' HP Q Q' HQ; split; intros [??]; split; by apply HP || by apply HQ.
 Qed.
-Global Instance iprop_and_proper :
-  Proper ((≡) ==> (≡) ==> (≡)) (@iprop_and M) := ne_proper_2 _.
-Global Instance iprop_or_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@iprop_or M).
+Global Instance uPred_and_proper :
+  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_and M) := ne_proper_2 _.
+Global Instance uPred_or_ne n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
 Proof.
  intros P P' HP Q Q' HQ; split; intros [?|?];
    first [by left; apply HP | by right; apply HQ].
 Qed.
-Global Instance iprop_or_proper :
-  Proper ((≡) ==> (≡) ==> (≡)) (@iprop_or M) := ne_proper_2 _.
-Global Instance iprop_impl_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@iprop_impl M).
+Global Instance uPred_or_proper :
+  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_or M) := ne_proper_2 _.
+Global Instance uPred_impl_ne n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_impl M).
 Proof.
  intros P P' HP Q Q' HQ; split; intros HPQ x' n'' ????; apply HQ,HPQ,HP; auto.
 Qed.
-Global Instance iprop_impl_proper :
-  Proper ((≡) ==> (≡) ==> (≡)) (@iprop_impl M) := ne_proper_2 _.
-Global Instance iprop_sep_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@iprop_sep M).
+Global Instance uPred_impl_proper :
+  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_impl M) := ne_proper_2 _.
+Global Instance uPred_sep_ne n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
 Proof.
  intros P P' HP Q Q' HQ x n' ? Hx'; split; intros (x1&x2&Hx&?&?);
    exists x1, x2; rewrite  Hx in Hx'; split_ands; try apply HP; try apply HQ;
    eauto using cmra_valid_op_l, cmra_valid_op_r.
 Qed.
-Global Instance iprop_sep_proper :
-  Proper ((≡) ==> (≡) ==> (≡)) (@iprop_sep M) := ne_proper_2 _.
-Global Instance iprop_wand_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@iprop_wand M).
+Global Instance uPred_sep_proper :
+  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_sep M) := ne_proper_2 _.
+Global Instance uPred_wand_ne n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_wand M).
 Proof.
  intros P P' HP Q Q' HQ x n' ??; split; intros HPQ x' n'' ???;
    apply HQ, HPQ, HP; eauto using cmra_valid_op_r.
 Qed.
-Global Instance iprop_wand_proper :
-  Proper ((≡) ==> (≡) ==> (≡)) (@iprop_wand M) := ne_proper_2 _.
-Global Instance iprop_eq_ne {A : cofeT} n :
-  Proper (dist n ==> dist n ==> dist n) (@iprop_eq M A).
+Global Instance uPred_wand_proper :
+  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_wand M) := ne_proper_2 _.
+Global Instance uPred_eq_ne {A : cofeT} n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
 Proof.
  intros x x' Hx y y' Hy z n'; split; intros; simpl in *.
  * by rewrite <-(dist_le _ _ _ _ Hx), <-(dist_le _ _ _ _ Hy) by auto.
  * by rewrite (dist_le _ _ _ _ Hx), (dist_le _ _ _ _ Hy) by auto.
 Qed.
-Global Instance iprop_eq_proper {A : cofeT} :
-  Proper ((≡) ==> (≡) ==> (≡)) (@iprop_eq M A) := ne_proper_2 _.
-Global Instance iprop_forall_ne {A : cofeT} :
-  Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_forall M A).
+Global Instance uPred_eq_proper {A : cofeT} :
+  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_eq M A) := ne_proper_2 _.
+Global Instance uPred_forall_ne {A : cofeT} :
+  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
 Proof. by intros n P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
-Global Instance iprop_forall_proper {A : cofeT} :
-  Proper (pointwise_relation _ (≡) ==> (≡)) (@iprop_forall M A).
+Global Instance uPred_forall_proper {A : cofeT} :
+  Proper (pointwise_relation _ (≡) ==> (≡)) (@uPred_forall M A).
 Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
-Global Instance iprop_exists_ne {A : cofeT} :
-  Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_exist M A).
+Global Instance uPred_exists_ne {A : cofeT} :
+  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
 Proof.
  by intros n P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
 Qed.
-Global Instance iprop_exist_proper {A : cofeT} :
-  Proper (pointwise_relation _ (≡) ==> (≡)) (@iprop_exist M A).
+Global Instance uPred_exist_proper {A : cofeT} :
+  Proper (pointwise_relation _ (≡) ==> (≡)) (@uPred_exist M A).
 Proof.
  by intros P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
 Qed.
-Global Instance iprop_later_contractive : Contractive (@iprop_later M).
+Global Instance uPred_later_contractive : Contractive (@uPred_later M).
 Proof.
  intros n P Q HPQ x [|n'] ??; simpl; [done|].
  apply HPQ; eauto using cmra_valid_S.
 Qed.
-Global Instance iprop_later_proper :
-  Proper ((≡) ==> (≡)) (@iprop_later M) := ne_proper _.
-Global Instance iprop_always_ne n: Proper (dist n ==> dist n) (@iprop_always M).
+Global Instance uPred_later_proper :
+  Proper ((≡) ==> (≡)) (@uPred_later M) := ne_proper _.
+Global Instance uPred_always_ne n: Proper (dist n ==> dist n) (@uPred_always M).
 Proof. intros P1 P2 HP x n'; split; apply HP; eauto using cmra_unit_valid. Qed.
-Global Instance iprop_always_proper :
-  Proper ((≡) ==> (≡)) (@iprop_always M) := ne_proper _.
-Global Instance iprop_own_ne n : Proper (dist n ==> dist n) (@iprop_own M).
+Global Instance uPred_always_proper :
+  Proper ((≡) ==> (≡)) (@uPred_always M) := ne_proper _.
+Global Instance uPred_own_ne n : Proper (dist n ==> dist n) (@uPred_own M).
 Proof.
  by intros a1 a2 Ha x n'; split; intros [a' ?]; exists a'; simpl; first
    [rewrite <-(dist_le _ _ _ _ Ha) by lia|rewrite (dist_le _ _ _ _ Ha) by lia].
 Qed.
-Global Instance iprop_own_proper :
-  Proper ((≡) ==> (≡)) (@iprop_own M) := ne_proper _.
+Global Instance uPred_own_proper :
+  Proper ((≡) ==> (≡)) (@uPred_own M) := ne_proper _.

 (** Introduction and elimination rules *)
-Lemma iprop_True_intro P : P ⊆ True%I.
+Lemma uPred_True_intro P : P ⊆ True%I.
 Proof. done. Qed.
-Lemma iprop_False_elim P : False%I ⊆ P.
+Lemma uPred_False_elim P : False%I ⊆ P.
 Proof. by intros x n ?. Qed.
-Lemma iprop_and_elim_l P Q : (P ∧ Q)%I ⊆ P.
+Lemma uPred_and_elim_l P Q : (P ∧ Q)%I ⊆ P.
 Proof. by intros x n ? [??]. Qed.
-Lemma iprop_and_elim_r P Q : (P ∧ Q)%I ⊆ Q.
+Lemma uPred_and_elim_r P Q : (P ∧ Q)%I ⊆ Q.
 Proof. by intros x n ? [??]. Qed.
-Lemma iprop_and_intro R P Q : R ⊆ P → R ⊆ Q → R ⊆ (P ∧ Q)%I.
+Lemma uPred_and_intro R P Q : R ⊆ P → R ⊆ Q → R ⊆ (P ∧ Q)%I.
 Proof. intros HP HQ x n ??; split. by apply HP. by apply HQ. Qed.
-Lemma iprop_or_intro_l P Q : P ⊆ (P ∨ Q)%I.
+Lemma uPred_or_intro_l P Q : P ⊆ (P ∨ Q)%I.
 Proof. by left. Qed.
-Lemma iprop_or_intro_r P Q : Q ⊆ (P ∨ Q)%I.
+Lemma uPred_or_intro_r P Q : Q ⊆ (P ∨ Q)%I.
 Proof. by right. Qed.
-Lemma iprop_or_elim R P Q : P ⊆ R → Q ⊆ R → (P ∨ Q)%I ⊆ R.
+Lemma uPred_or_elim R P Q : P ⊆ R → Q ⊆ R → (P ∨ Q)%I ⊆ R.
 Proof. intros HP HQ x n ? [?|?]. by apply HP. by apply HQ. Qed.
-Lemma iprop_impl_intro P Q R : (R ∧ P)%I ⊆ Q → R ⊆ (P → Q)%I.
+Lemma uPred_impl_intro P Q R : (R ∧ P)%I ⊆ Q → R ⊆ (P → Q)%I.
 Proof.
-  intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using iprop_weaken.
+  intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using uPred_weaken.
 Qed.
-Lemma iprop_impl_elim P Q : ((P → Q) ∧ P)%I ⊆ Q.
+Lemma uPred_impl_elim P Q : ((P → Q) ∧ P)%I ⊆ Q.
 Proof. by intros x n ? [HQ HP]; apply HQ. Qed.
-Lemma iprop_forall_intro P `(Q: A → iProp M): (∀ a, P ⊆ Q a) → P ⊆ (∀ a, Q a)%I.
+Lemma uPred_forall_intro P `(Q: A → uPred M): (∀ a, P ⊆ Q a) → P ⊆ (∀ a, Q a)%I.
 Proof. by intros HPQ x n ?? a; apply HPQ. Qed.
-Lemma iprop_forall_elim `(P : A → iProp M) a : (∀ a, P a)%I ⊆ P a.
+Lemma uPred_forall_elim `(P : A → uPred M) a : (∀ a, P a)%I ⊆ P a.
 Proof. intros x n ? HP; apply HP. Qed.
-Lemma iprop_exist_intro `(P : A → iProp M) a : P a ⊆ (∃ a, P a)%I.
+Lemma uPred_exist_intro `(P : A → uPred M) a : P a ⊆ (∃ a, P a)%I.
 Proof. by intros x n ??; exists a. Qed.
-Lemma iprop_exist_elim `(P : A → iProp M) Q : (∀ a, P a ⊆ Q) → (∃ a, P a)%I ⊆ Q.
+Lemma uPred_exist_elim `(P : A → uPred M) Q : (∀ a, P a ⊆ Q) → (∃ a, P a)%I ⊆ Q.
 Proof. by intros HPQ x n ? [a ?]; apply HPQ with a. Qed.

 (* BI connectives *)
-Lemma iprop_sep_elim_l P Q : (P ★ Q)%I ⊆ P.
+Lemma uPred_sep_elim_l P Q : (P ★ Q)%I ⊆ P.
 Proof.
  intros x n Hvalid (x1&x2&Hx&?&?); rewrite Hx in Hvalid |- *.
-  by apply iprop_weaken with x1 n; auto using ra_included_l.
+  by apply uPred_weaken with x1 n; auto using ra_included_l.
 Qed.
-Global Instance iprop_sep_left_id : LeftId (≡) True%I (@iprop_sep M).
+Global Instance uPred_sep_left_id : LeftId (≡) True%I (@uPred_sep M).
 Proof.
  intros P x n Hvalid; split.
  * intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
-    apply iprop_weaken with x2 n; auto using ra_included_r.
+    apply uPred_weaken with x2 n; auto using ra_included_r.
  * by intros ?; exists (unit x), x; rewrite ra_unit_l.
 Qed. 
-Global Instance iprop_sep_commutative : Commutative (≡) (@iprop_sep M).
+Global Instance uPred_sep_commutative : Commutative (≡) (@uPred_sep M).
 Proof.
  by intros P Q x n ?; split;
    intros (x1&x2&?&?&?); exists x2, x1; rewrite (commutative op).
 Qed.
-Global Instance iprop_sep_associative : Associative (≡) (@iprop_sep M).
+Global Instance uPred_sep_associative : Associative (≡) (@uPred_sep M).
 Proof.
  intros P Q R x n ?; split.
  * intros (x1&x2&Hx&?&y1&y2&Hy&?&?); exists (x1 ⋅ y1), y2; split_ands; auto.
@@ -337,65 +337,65 @@ Proof.
    + by rewrite (associative op), <-Hy, <-Hx.
    + by exists y2, x2.
 Qed.
-Lemma iprop_wand_intro P Q R : (R ★ P)%I ⊆ Q → R ⊆ (P -★ Q)%I.
+Lemma uPred_wand_intro P Q R : (R ★ P)%I ⊆ Q → R ⊆ (P -★ Q)%I.
 Proof.
  intros HPQ x n ?? x' n' ???; apply HPQ; auto.
  exists x, x'; split_ands; auto.
-  eapply iprop_weaken with x n; eauto using cmra_valid_op_l.
+  eapply uPred_weaken with x n; eauto using cmra_valid_op_l.
 Qed.
-Lemma iprop_wand_elim P Q : ((P -★ Q) ★ P)%I ⊆ Q.
+Lemma uPred_wand_elim P Q : ((P -★ Q) ★ P)%I ⊆ Q.
 Proof.
  by intros x n Hvalid (x1&x2&Hx&HPQ&?); rewrite Hx in Hvalid |- *; apply HPQ.
 Qed.
-Lemma iprop_sep_or P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I.
+Lemma uPred_sep_or P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I.
 Proof.
  split; [by intros (x1&x2&Hx&[?|?]&?); [left|right]; exists x1, x2|].
  intros [(x1&x2&Hx&?&?)|(x1&x2&Hx&?&?)]; exists x1, x2; split_ands;
    first [by left | by right | done].