Put logic stuff in module.

07f5e2fe · Robbert Krebbers · b3c56843 · 07f5e2fe
Commit 07f5e2fe authored Dec 15, 2015 by Robbert Krebbers
--- a/modures/logic.v
+++ b/modures/logic.v
@@ -203,281 +203,275 @@ Infix "↔" := uPred_iff : uPred_scope.

 Class TimelessP {M} (P : uPred M) := timelessP x n : ✓{1} x → P 1 x → P n x.

-Section logic.
+Module uPred. Section uPred_logic.
 Context {M : cmraT}.
 Implicit Types P Q : uPred M.
 Implicit Types A : Type.

-Global Instance uPred_preorder : PreOrder ((⊆) : relation (uPred M)).
+Global Instance: PreOrder ((⊆) : relation (uPred M)).
 Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
-Global Instance uPred_antisym : AntiSymmetric (≡) ((⊆) : relation (uPred M)).
+Global Instance: AntiSymmetric (≡) ((⊆) : relation (uPred M)).
 Proof. intros P Q HPQ HQP; split; auto. Qed.
-Lemma uPred_equiv_spec P Q : P ≡ Q ↔ P ⊆ Q ∧ Q ⊆ P.
+Lemma equiv_spec P Q : P ≡ Q ↔ P ⊆ Q ∧ Q ⊆ P.
 Proof.
  split; [|by intros [??]; apply (anti_symmetric (⊆))].
  intros HPQ; split; intros x i; apply HPQ.
 Qed.
-Global Instance uPred_entails_proper :
+Global Instance entails_proper :
  Proper ((≡) ==> (≡) ==> iff) ((⊆) : relation (uPred M)).
 Proof.
-  intros P1 P2 HP Q1 Q2 HQ; rewrite uPred_equiv_spec in HP, HQ; split; intros.
+  intros P1 P2 HP Q1 Q2 HQ; rewrite equiv_spec in HP, HQ; split; intros.
  * by rewrite (proj2 HP), <-(proj1 HQ).
  * by rewrite (proj1 HP), <-(proj2 HQ).
 Qed.

 (** Non-expansiveness and setoid morphisms *)
-Global Instance uPred_const_proper : Proper (iff ==> (≡)) (@uPred_const M).
+Global Instance const_proper : Proper (iff ==> (≡)) (@uPred_const M).
 Proof. by intros P Q HPQ ? [|n] ?; try apply HPQ. Qed.
-Global Instance uPred_and_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
+Global Instance 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 uPred_and_proper :
+Global Instance 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).
+Global Instance 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 uPred_or_proper :
+Global Instance or_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_or M) := ne_proper_2 _.
-Global Instance uPred_impl_ne n :
+Global Instance 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 uPred_impl_proper :
+Global Instance 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).
+Global Instance 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 uPred_sep_proper :
+Global Instance sep_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_sep M) := ne_proper_2 _.
-Global Instance uPred_wand_ne n :
+Global Instance 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 uPred_wand_proper :
+Global Instance wand_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_wand M) := ne_proper_2 _.
-Global Instance uPred_eq_ne (A : cofeT) n :
+Global Instance 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 uPred_eq_proper (A : cofeT) :
+Global Instance eq_proper (A : cofeT) :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_eq M A) := ne_proper_2 _.
-Global Instance uPred_forall_ne A :
+Global Instance forall_ne A :
  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 uPred_forall_proper A :
+Global Instance forall_proper A :
  Proper (pointwise_relation _ (≡) ==> (≡)) (@uPred_forall M A).
 Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
-Global Instance uPred_exists_ne A :
+Global Instance exists_ne A :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
 Proof.
  by intros n P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
 Qed.
-Global Instance uPred_exist_proper A :
+Global Instance exist_proper A :
  Proper (pointwise_relation _ (≡) ==> (≡)) (@uPred_exist M A).
 Proof.
  by intros P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
 Qed.
-Global Instance uPred_later_contractive : Contractive (@uPred_later M).
+Global Instance 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 uPred_later_proper :
+Global Instance later_proper :
  Proper ((≡) ==> (≡)) (@uPred_later M) := ne_proper _.
-Global Instance uPred_always_ne n: Proper (dist n ==> dist n) (@uPred_always M).
+Global Instance 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 uPred_always_proper :
+Global Instance always_proper :
  Proper ((≡) ==> (≡)) (@uPred_always M) := ne_proper _.
-Global Instance uPred_own_ne n : Proper (dist n ==> dist n) (@uPred_own M).
+Global Instance 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 uPred_own_proper :
-  Proper ((≡) ==> (≡)) (@uPred_own M) := ne_proper _.
-Global Instance uPred_iff_ne n :
-  Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
+Global Instance own_proper : Proper ((≡) ==> (≡)) (@uPred_own M) := ne_proper _.
+Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
 Proof. unfold uPred_iff; solve_proper. Qed.
-Global Instance uPred_iff_proper :
+Global Instance iff_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_iff M) := ne_proper_2 _.

 (** Introduction and elimination rules *)
-Lemma uPred_const_intro P (Q : Prop) : Q → P ⊆ uPred_const Q.
+Lemma const_intro P (Q : Prop) : Q → P ⊆ uPred_const Q.
 Proof. by intros ?? [|?]. Qed.
-Lemma uPred_const_elim (P : Prop) Q R : (P → Q ⊆ R) → (Q ∧ uPred_const P)%I ⊆ R.
+Lemma const_elim (P : Prop) Q R : (P → Q ⊆ R) → (Q ∧ uPred_const P)%I ⊆ R.
 Proof. intros HR x [|n] ? [??]; [|apply HR]; auto. Qed.
-Lemma uPred_True_intro P : P ⊆ True%I.
-Proof. by apply uPred_const_intro. Qed.
-Lemma uPred_False_elim P : False%I ⊆ P.
+Lemma True_intro P : P ⊆ True%I.
+Proof. by apply const_intro. Qed.
+Lemma False_elim P : False%I ⊆ P.
 Proof. by intros x [|n] ?. Qed.
-Lemma uPred_and_elim_l P Q : (P ∧ Q)%I ⊆ P.
+Lemma and_elim_l P Q : (P ∧ Q)%I ⊆ P.
 Proof. by intros x n ? [??]. Qed.
-Lemma uPred_and_elim_r P Q : (P ∧ Q)%I ⊆ Q.
+Lemma and_elim_r P Q : (P ∧ Q)%I ⊆ Q.
 Proof. by intros x n ? [??]. Qed.
-Lemma uPred_and_intro P Q R : P ⊆ Q → P ⊆ R → P ⊆ (Q ∧ R)%I.
+Lemma and_intro P Q R : P ⊆ Q → P ⊆ R → P ⊆ (Q ∧ R)%I.
 Proof. intros HQ HR x n ??; split; auto. Qed.
-Lemma uPred_or_intro_l P Q R : P ⊆ Q → P ⊆ (Q ∨ R)%I.
+Lemma or_intro_l P Q R : P ⊆ Q → P ⊆ (Q ∨ R)%I.
 Proof. intros HQ x n ??; left; auto. Qed.
-Lemma uPred_or_intro_r P Q R : P ⊆ R → P ⊆ (Q ∨ R)%I.
+Lemma or_intro_r P Q R : P ⊆ R → P ⊆ (Q ∨ R)%I.
 Proof. intros HR x n ??; right; auto. Qed.
-Lemma uPred_or_elim R P Q : P ⊆ R → Q ⊆ R → (P ∨ Q)%I ⊆ R.
+Lemma 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 uPred_impl_intro P Q R : (R ∧ P)%I ⊆ Q → R ⊆ (P → Q)%I.
+Lemma 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 uPred_weaken.
 Qed.
-Lemma uPred_impl_elim P Q R : P ⊆ (Q → R)%I → P ⊆ Q → P ⊆ R.
+Lemma impl_elim P Q R : P ⊆ (Q → R)%I → P ⊆ Q → P ⊆ R.
 Proof. by intros HP HP' x n ??; apply HP with x n, HP'. Qed.
-Lemma uPred_forall_intro P `(Q: A → uPred M): (∀ a, P ⊆ Q a) → P ⊆ (∀ a, Q a)%I.
+Lemma 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 uPred_forall_elim `(P : A → uPred M) a : (∀ a, P a)%I ⊆ P a.
+Lemma forall_elim `(P : A → uPred M) a : (∀ a, P a)%I ⊆ P a.
 Proof. intros x n ? HP; apply HP. Qed.
-Lemma uPred_exist_intro `(P : A → uPred M) a : P a ⊆ (∃ a, P a)%I.
+Lemma exist_intro `(P : A → uPred M) a : P a ⊆ (∃ a, P a)%I.
 Proof. by intros x [|n] ??; [done|exists a]. Qed.
-Lemma uPred_exist_elim `(P : A → uPred M) Q : (∀ a, P a ⊆ Q) → (∃ a, P a)%I ⊆ Q.
+Lemma exist_elim `(P : A → uPred M) Q : (∀ a, P a ⊆ Q) → (∃ a, P a)%I ⊆ Q.
 Proof. by intros HPQ x [|n] ?; [|intros [a ?]; apply HPQ with a]. Qed.
-Lemma uPred_eq_refl {A : cofeT} (a : A) P : P ⊆ (a ≡ a)%I.
+Lemma eq_refl {A : cofeT} (a : A) P : P ⊆ (a ≡ a)%I.
 Proof. by intros x n ??; simpl. Qed.
-Lemma uPred_eq_rewrite {A : cofeT} P (Q : A → uPred M)
+Lemma eq_rewrite {A : cofeT} P (Q : A → uPred M)
    `{HQ : !∀ n, Proper (dist n ==> dist n) Q} a b :
  P ⊆ (a ≡ b)%I → P ⊆ Q a → P ⊆ Q b.
 Proof.
  intros Hab Ha x n ??; apply HQ with n a; auto. by symmetry; apply Hab with x.
 Qed.
-Lemma uPred_eq_equiv `{Empty M, !RAEmpty M} {A : cofeT} (a b : A) :
+Lemma eq_equiv `{Empty M, !RAEmpty M} {A : cofeT} (a b : A) :
  True%I ⊆ (a ≡ b : uPred M)%I → a ≡ b.
 Proof.
  intros Hab; apply equiv_dist; intros n; apply Hab with ∅.
  * apply cmra_valid_validN, ra_empty_valid.
  * by destruct n.
 Qed.
-Lemma uPred_iff_equiv P Q : True%I ⊆ (P ↔ Q)%I → P ≡ Q.
+Lemma iff_equiv P Q : True%I ⊆ (P ↔ Q)%I → P ≡ Q.
 Proof. by intros HPQ x [|n] ?; [|split; intros; apply HPQ with x (S n)]. Qed.

 (* Derived logical stuff *)
-Lemma uPred_and_elim_l' P Q R : P ⊆ R → (P ∧ Q)%I ⊆ R.
-Proof. by rewrite uPred_and_elim_l. Qed.
-Lemma uPred_and_elim_r' P Q R : Q ⊆ R → (P ∧ Q)%I ⊆ R.
-Proof. by rewrite uPred_and_elim_r. Qed.
+Lemma and_elim_l' P Q R : P ⊆ R → (P ∧ Q)%I ⊆ R.
+Proof. by rewrite and_elim_l. Qed.
+Lemma and_elim_r' P Q R : Q ⊆ R → (P ∧ Q)%I ⊆ R.
+Proof. by rewrite and_elim_r. Qed.

-Hint Resolve uPred_or_elim uPred_or_intro_l uPred_or_intro_r.
-Hint Resolve uPred_and_intro uPred_and_elim_l' uPred_and_elim_r'.
-Hint Immediate uPred_True_intro uPred_False_elim.
+Hint Resolve or_elim or_intro_l or_intro_r.
+Hint Resolve and_intro and_elim_l' and_elim_r'.
+Hint Immediate True_intro False_elim.

-Global Instance uPred_and_idem : Idempotent (≡) (@uPred_and M).
+Global Instance and_idem : Idempotent (≡) (@uPred_and M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_or_idem : Idempotent (≡) (@uPred_or M).
+Global Instance or_idem : Idempotent (≡) (@uPred_or M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_and_comm : Commutative (≡) (@uPred_and M).
+Global Instance and_comm : Commutative (≡) (@uPred_and M).
 Proof. intros P Q; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_True_and : LeftId (≡) True%I (@uPred_and M).
+Global Instance True_and : LeftId (≡) True%I (@uPred_and M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_and_True : RightId (≡) True%I (@uPred_and M).
+Global Instance and_True : RightId (≡) True%I (@uPred_and M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_False_and : LeftAbsorb (≡) False%I (@uPred_and M).
+Global Instance False_and : LeftAbsorb (≡) False%I (@uPred_and M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_and_False : RightAbsorb (≡) False%I (@uPred_and M).
+Global Instance and_False : RightAbsorb (≡) False%I (@uPred_and M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_True_or : LeftAbsorb (≡) True%I (@uPred_or M).
+Global Instance True_or : LeftAbsorb (≡) True%I (@uPred_or M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_or_True : RightAbsorb (≡) True%I (@uPred_or M).
+Global Instance or_True : RightAbsorb (≡) True%I (@uPred_or M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_False_or : LeftId (≡) False%I (@uPred_or M).
+Global Instance False_or : LeftId (≡) False%I (@uPred_or M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_or_False : RightId (≡) False%I (@uPred_or M).
+Global Instance or_False : RightId (≡) False%I (@uPred_or M).
 Proof. intros P; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_and_assoc : Associative (≡) (@uPred_and M).
+Global Instance and_assoc : Associative (≡) (@uPred_and M).
 Proof. intros P Q R; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_or_comm : Commutative (≡) (@uPred_or M).
+Global Instance or_comm : Commutative (≡) (@uPred_or M).
 Proof. intros P Q; apply (anti_symmetric (⊆)); auto. Qed.
-Global Instance uPred_or_assoc : Associative (≡) (@uPred_or M).
+Global Instance or_assoc : Associative (≡) (@uPred_or M).
 Proof. intros P Q R; apply (anti_symmetric (⊆)); auto. Qed.

-Lemma uPred_const_mono (P Q: Prop) : (P → Q) → uPred_const P ⊆ @uPred_const M Q.
+Lemma const_mono (P Q: Prop) : (P → Q) → uPred_const P ⊆ @uPred_const M Q.
 Proof.
  intros; rewrite <-(left_id True%I _ (uPred_const P)).
-  eauto using uPred_const_elim, uPred_const_intro.
+  eauto using const_elim, const_intro.
 Qed.
-Lemma uPred_and_mono P P' Q Q' : P ⊆ Q → P' ⊆ Q' → (P ∧ P')%I ⊆ (Q ∧ Q')%I.
+Lemma and_mono P P' Q Q' : P ⊆ Q → P' ⊆ Q' → (P ∧ P')%I ⊆ (Q ∧ Q')%I.
 Proof. auto. Qed.
-Lemma uPred_or_mono P P' Q Q' : P ⊆ Q → P' ⊆ Q' → (P ∨ P')%I ⊆ (Q ∨ Q')%I.
+Lemma or_mono P P' Q Q' : P ⊆ Q → P' ⊆ Q' → (P ∨ P')%I ⊆ (Q ∨ Q')%I.
 Proof. auto. Qed.
-Lemma uPred_impl_mono P P' Q Q' : Q ⊆ P → P' ⊆ Q' → (P → P')%I ⊆ (Q → Q')%I.
+Lemma impl_mono P P' Q Q' : Q ⊆ P → P' ⊆ Q' → (P → P')%I ⊆ (Q → Q')%I.
 Proof.
-  intros HP HQ'; apply uPred_impl_intro; rewrite <-HQ'.
-  transitivity ((P → P') ∧ P)%I; eauto using uPred_impl_elim.
+  intros HP HQ'; apply impl_intro; rewrite <-HQ'.
+  transitivity ((P → P') ∧ P)%I; eauto using impl_elim.
 Qed.
-Lemma uPred_forall_mono {A} (P Q : A → uPred M) :
+Lemma forall_mono {A} (P Q : A → uPred M) :
  (∀ a, P a ⊆ Q a) → (∀ a, P a)%I ⊆ (∀ a, Q a)%I.
 Proof.
-  intros HPQ. apply uPred_forall_intro; intros a; rewrite <-(HPQ a).
-  apply uPred_forall_elim.
+  intros HPQ. apply forall_intro; intros a; rewrite <-(HPQ a).
+  apply forall_elim.
 Qed.
-Lemma uPred_exist_mono {A} (P Q : A → uPred M) :
+Lemma exist_mono {A} (P Q : A → uPred M) :
  (∀ a, P a ⊆ Q a) → (∃ a, P a)%I ⊆ (∃ a, Q a)%I.
 Proof.
-  intros HPQ. apply uPred_exist_elim; intros a; rewrite (HPQ a).
-  apply uPred_exist_intro.
+  intros HPQ. apply exist_elim; intros a; rewrite (HPQ a); apply exist_intro.
 Qed.

-Global Instance uPred_const_mono' : Proper (impl ==> (⊆)) (@uPred_const M).
-Proof. intros P Q; apply uPred_const_mono. Qed.
-Global Instance uPred_and_mono' : Proper ((⊆) ==> (⊆) ==> (⊆)) (@uPred_and M).
-Proof. by intros P P' HP Q Q' HQ; apply uPred_and_mono. Qed.
-Global Instance uPred_or_mono' : Proper ((⊆) ==> (⊆) ==> (⊆)) (@uPred_or M).
-Proof. by intros P P' HP Q Q' HQ; apply uPred_or_mono. Qed.
-Global Instance uPred_impl_mono' :
+Global Instance const_mono' : Proper (impl ==> (⊆)) (@uPred_const M).
+Proof. intros P Q; apply const_mono. Qed.
+Global Instance and_mono' : Proper ((⊆) ==> (⊆) ==> (⊆)) (@uPred_and M).
+Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
+Global Instance or_mono' : Proper ((⊆) ==> (⊆) ==> (⊆)) (@uPred_or M).
+Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
+Global Instance impl_mono' :
  Proper (flip (⊆) ==> (⊆) ==> (⊆)) (@uPred_impl M).
-Proof. by intros P P' HP Q Q' HQ; apply uPred_impl_mono. Qed.
-Global Instance uPred_forall_mono' A :
+Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
+Global Instance forall_mono' A :
  Proper (pointwise_relation _ (⊆) ==> (⊆)) (@uPred_forall M A).
-Proof. intros P1 P2; apply uPred_forall_mono. Qed.
-Global Instance uPred_exist_mono' A :
+Proof. intros P1 P2; apply forall_mono. Qed.
+Global Instance exist_mono' A :
  Proper (pointwise_relation _ (⊆) ==> (⊆)) (@uPred_exist M A).
-Proof. intros P1 P2; apply uPred_exist_mono. Qed.
+Proof. intros P1 P2; apply exist_mono. Qed.

-Lemma uPred_equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊆ (a ≡ b)%I.
-Proof. intros ->; apply uPred_eq_refl. Qed.
-Lemma uPred_eq_sym {A : cofeT} (a b : A) : (a ≡ b)%I ⊆ (b ≡ a : uPred M)%I.
+Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊆ (a ≡ b)%I.
+Proof. intros ->; apply eq_refl. Qed.
+Lemma eq_sym {A : cofeT} (a b : A) : (a ≡ b)%I ⊆ (b ≡ a : uPred M)%I.
 Proof.
-  refine (uPred_eq_rewrite _ (λ b, b ≡ a)%I a b _ _); auto using uPred_eq_refl.
+  refine (eq_rewrite _ (λ b, b ≡ a)%I a b _ _); auto using eq_refl.
  intros n; solve_proper.
 Qed.

 (* BI connectives *)
-Lemma uPred_sep_mono P P' Q Q' : P ⊆ Q → P' ⊆ Q' → (P ★ P')%I ⊆ (Q ★ Q')%I.
+Lemma sep_mono P P' Q Q' : P ⊆ Q → P' ⊆ Q' → (P ★ P')%I ⊆ (Q ★ Q')%I.
 Proof.
  intros HQ HQ' x n' Hx' (x1&x2&Hx&?&?); exists x1, x2;
    rewrite Hx in Hx'; eauto 7 using cmra_valid_op_l, cmra_valid_op_r.
 Qed.
-Global Instance uPred_True_sep : LeftId (≡) True%I (@uPred_sep M).
+Global Instance True_sep : LeftId (≡) True%I (@uPred_sep M).
 Proof.
  intros P x n Hvalid; split.
  * intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
    eauto using uPred_weaken, ra_included_r.
  * by destruct n as [|n]; [|intros ?; exists (unit x), x; rewrite ra_unit_l].
 Qed. 
-Global Instance uPred_sep_commutative : Commutative (≡) (@uPred_sep M).
+Global Instance 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 uPred_sep_associative : Associative (≡) (@uPred_sep M).
+Global Instance 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.
@@ -487,23 +481,23 @@ Proof.
    + by rewrite (associative op), <-Hy, <-Hx.
    + by exists y2, x2.
 Qed.
-Lemma uPred_wand_intro P Q R : (R ★ P)%I ⊆ Q → R ⊆ (P -★ Q)%I.
+Lemma 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 uPred_weaken with x n; eauto using cmra_valid_op_l.
 Qed.
-Lemma uPred_wand_elim P Q : ((P -★ Q) ★ P)%I ⊆ Q.
+Lemma 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 uPred_or_sep_distr P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I.
+Lemma or_sep_distr 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].
 Qed.
-Lemma uPred_exist_sep_distr `(P : A → uPred M) Q :
+Lemma exist_sep_distr `(P : A → uPred M) Q :
  ((∃ b, P b) ★ Q)%I ≡ (∃ b, P b ★ Q)%I.
 Proof.
  intros x [|n]; [done|split; [by intros (x1&x2&Hx&[a ?]&?); exists a,x1,x2|]].
@@ -511,63 +505,62 @@ Proof.
 Qed.

 (* Derived BI Stuff *)
-Hint Resolve uPred_sep_mono.
-Global Instance uPred_sep_mono' : Proper ((⊆) ==> (⊆) ==> (⊆)) (@uPred_sep M).
-Proof. by intros P P' HP Q Q' HQ; apply uPred_sep_mono. Qed.
-Lemma uPred_wand_mono P P' Q Q' : Q ⊆ P → P' ⊆ Q' → (P -★ P')%I ⊆ (Q -★ Q')%I.
+Hint Resolve sep_mono.
+Global Instance sep_mono' : Proper ((⊆) ==> (⊆) ==> (⊆)) (@uPred_sep M).
+Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
+Lemma wand_mono P P' Q Q' : Q ⊆ P → P' ⊆ Q' → (P -★ P')%I ⊆ (Q -★ Q')%I.
 Proof.
-  intros HP HQ; apply uPred_wand_intro; rewrite HP, <-HQ; apply uPred_wand_elim.
+  intros HP HQ; apply wand_intro; rewrite HP, <-HQ; apply wand_elim.
 Qed.
-Global Instance uPred_wand_mono' :
-  Proper (flip (⊆) ==> (⊆) ==> (⊆)) (@uPred_wand M).
-Proof. by intros P P' HP Q Q' HQ; apply uPred_wand_mono. Qed.
+Global Instance wand_mono' : Proper (flip (⊆) ==> (⊆) ==> (⊆)) (@uPred_wand M).
+Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.

-Global Instance uPred_sep_True : RightId (≡) True%I (@uPred_sep M).
+Global Instance sep_True : RightId (≡) True%I (@uPred_sep M).
 Proof. by intros P; rewrite (commutative _), (left_id _ _). Qed.
-Lemma uPred_sep_elim_l P Q R : P ⊆ R → (P ★ Q)%I ⊆ R.
-Proof. by intros HR; rewrite <-(right_id _ (★) R)%I, HR, (uPred_True_intro Q). Qed.
-Lemma uPred_sep_elim_r P Q : (P ★ Q)%I ⊆ Q.
-Proof. by rewrite (commutative (★))%I; apply uPred_sep_elim_l. Qed.
-Hint Resolve uPred_sep_elim_l uPred_sep_elim_r.
-Lemma uPred_sep_and P Q : (P ★ Q)%I ⊆ (P ∧ Q)%I.
+Lemma sep_elim_l P Q R : P ⊆ R → (P ★ Q)%I ⊆ R.
+Proof. by intros HR; rewrite <-(right_id _ (★) R)%I, HR, (True_intro Q). Qed.
+Lemma sep_elim_r P Q : (P ★ Q)%I ⊆ Q.
+Proof. by rewrite (commutative (★))%I; apply sep_elim_l. Qed.
+Hint Resolve sep_elim_l sep_elim_r.
+Lemma sep_and P Q : (P ★ Q)%I ⊆ (P ∧ Q)%I.
 Proof. auto. Qed.
-Global Instance uPred_sep_False : LeftAbsorb (≡) False%I (@uPred_sep M).
+Global Instance sep_False : LeftAbsorb (≡) False%I (@uPred_sep M).
 Proof. intros P; apply (anti_symmetric _); auto. Qed.
-Global Instance uPred_False_sep : RightAbsorb (≡) False%I (@uPred_sep M).
+Global Instance False_sep : RightAbsorb (≡) False%I (@uPred_sep M).
 Proof. intros P; apply (anti_symmetric _); auto. Qed.
-Lemma uPred_impl_wand P Q : (P → Q)%I ⊆ (P -★ Q)%I.
-Proof. apply uPred_wand_intro, uPred_impl_elim with P; auto. Qed.
-Lemma uPred_and_sep_distr P Q R : ((P ∧ Q) ★ R)%I ⊆ ((P ★ R) ∧ (Q ★ R))%I.
+Lemma impl_wand P Q : (P → Q)%I ⊆ (P -★ Q)%I.
+Proof. apply wand_intro, impl_elim with P; auto. Qed.
+Lemma and_sep_distr P Q R : ((P ∧ Q) ★ R)%I ⊆ ((P ★ R) ∧ (Q ★ R))%I.
 Proof. auto. Qed.
-Lemma uPred_forall_sep_distr `(P : A → uPred M) Q :
+Lemma forall_sep_distr `(P : A → uPred M) Q :
  ((∀ a, P a) ★ Q)%I ⊆ (∀ a, P a ★ Q)%I.
-Proof. by apply uPred_forall_intro; intros a; rewrite uPred_forall_elim. Qed.
+Proof. by apply forall_intro; intros a; rewrite forall_elim. Qed.

 (* Later *)
-Lemma uPred_later_mono P Q : P ⊆ Q → (▷ P)%I ⊆ (▷ Q)%I.
+Lemma later_mono P Q : P ⊆ Q → (▷ P)%I ⊆ (▷ Q)%I.
 Proof. intros HP x [|n] ??; [done|apply HP; auto using cmra_valid_S]. Qed.
-Lemma uPred_later_intro P : P ⊆ (▷ P)%I.
+Lemma later_intro P : P ⊆ (▷ P)%I.
 Proof.
  intros x [|n] ??; simpl in *; [done|].
  apply uPred_weaken with x (S n); auto using cmra_valid_S.
 Qed.
-Lemma uPred_lub P : (▷ P → P)%I ⊆ P.
+Lemma lub P : (▷ P → P)%I ⊆ P.
 Proof.
  intros x n ? HP; induction n as [|n IH]; [by apply HP|].
  apply HP, IH, uPred_weaken with x (S n); eauto using cmra_valid_S.
 Qed.
-Lemma uPred_later_and P Q : (▷ (P ∧ Q))%I ≡ (▷ P ∧ ▷ Q)%I.
+Lemma later_and P Q : (▷ (P ∧ Q))%I ≡ (▷ P ∧ ▷ Q)%I.
 Proof. by intros x [|n]; split. Qed.
-Lemma uPred_later_or P Q : (▷ (P ∨ Q))%I ≡ (▷ P ∨ ▷ Q)%I.
+Lemma later_or P Q : (▷ (P ∨ Q))%I ≡ (▷ P ∨ ▷ Q)%I.
 Proof. intros x [|n]; simpl; tauto. Qed.