AffineBI -> BiAffine, PositiveBI -> BiPositive.

5524871e · Jacques-Henri Jourdan · d831f1e2 · 5524871e · 5524871e · 5524871e
Commit 5524871e authored Dec 04, 2017 by Jacques-Henri Jourdan
--- a/theories/base_logic/derived.v
+++ b/theories/base_logic/derived.v
@@ -32,7 +32,7 @@ Proof.
 Qed.

 (* Affine *)
-Global Instance uPred_affine : AffineBI (uPredI M) | 0.
+Global Instance uPred_affine : BiAffine (uPredI M) | 0.
 Proof. intros P. rewrite /Affine. by apply bi.pure_intro. Qed.

 (* Own and valid derived *)
@@ -131,6 +131,6 @@ Proof. split; [split; try apply _|]. apply ownM_op. apply ownM_unit'. Qed.
 End derived.

 (* Also add this to the global hint database, otherwise [eauto] won't work for
-many lemmas that have [AffineBI] as a premise. *)
+many lemmas that have [BiAffine] as a premise. *)
 Hint Immediate uPred_affine.
 End uPred.
--- a/theories/bi/big_op.v
+++ b/theories/bi/big_op.v
@@ -55,7 +55,7 @@ Section sep_list.

  Lemma big_sepL_nil Φ : ([∗ list] k↦y ∈ nil, Φ k y) ⊣⊢ emp.
  Proof. done. Qed.
-  Lemma big_sepL_nil' `{AffineBI PROP} P Φ : P ⊢ [∗ list] k↦y ∈ nil, Φ k y.
+  Lemma big_sepL_nil' `{BiAffine PROP} P Φ : P ⊢ [∗ list] k↦y ∈ nil, Φ k y.
  Proof. apply (affine _). Qed.
  Lemma big_sepL_cons Φ x l :
    ([∗ list] k↦y ∈ x :: l, Φ k y) ⊣⊢ Φ 0 x ∗ [∗ list] k↦y ∈ l, Φ (S k) y.
@@ -75,7 +75,7 @@ Section sep_list.
    (∀ k y, l !! k = Some y → Φ k y ⊣⊢ Ψ k y) →
    ([∗ list] k ↦ y ∈ l, Φ k y) ⊣⊢ ([∗ list] k ↦ y ∈ l, Ψ k y).
  Proof. apply big_opL_proper. Qed.
-  Lemma big_sepL_submseteq `{AffineBI PROP} (Φ : A → PROP) l1 l2 :
+  Lemma big_sepL_submseteq `{BiAffine PROP} (Φ : A → PROP) l1 l2 :
    l1 ⊆+ l2 → ([∗ list] y ∈ l2, Φ y) ⊢ [∗ list] y ∈ l1, Φ y.
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_sepL_app sep_elim_l.
@@ -125,16 +125,16 @@ Section sep_list.
    ⊢ ([∗ list] k↦x ∈ l, Φ k x) ∧ ([∗ list] k↦x ∈ l, Ψ k x).
  Proof. auto using and_intro, big_sepL_mono, and_elim_l, and_elim_r. Qed.

-  Lemma big_sepL_plainly `{AffineBI PROP} Φ l :
+  Lemma big_sepL_plainly `{BiAffine PROP} Φ l :
    bi_plainly ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ [∗ list] k↦x ∈ l, bi_plainly (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

-  Lemma big_sepL_persistently `{AffineBI PROP} Φ l :
+  Lemma big_sepL_persistently `{BiAffine PROP} Φ l :
    bi_persistently ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢
    [∗ list] k↦x ∈ l, bi_persistently (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

-  Lemma big_sepL_forall `{AffineBI PROP} Φ l :
+  Lemma big_sepL_forall `{BiAffine PROP} Φ l :
    (∀ k x, Persistent (Φ k x)) →
    ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x).
  Proof.
@@ -287,7 +287,7 @@ Section and_list.
    [∧ list] k↦x ∈ l, bi_persistently (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

-  Lemma big_andL_forall `{AffineBI PROP} Φ l :
+  Lemma big_andL_forall `{BiAffine PROP} Φ l :
    ([∧ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x).
  Proof.
    apply (anti_symm _).
@@ -328,7 +328,7 @@ Section gmap.
    (∀ k x, m !! k = Some x → Φ k x ⊣⊢ Ψ k x) →
    ([∗ map] k ↦ x ∈ m, Φ k x) ⊣⊢ ([∗ map] k ↦ x ∈ m, Ψ k x).
  Proof. apply big_opM_proper. Qed.
-  Lemma big_sepM_subseteq `{AffineBI PROP} Φ m1 m2 :
+  Lemma big_sepM_subseteq `{BiAffine PROP} Φ m1 m2 :
    m2 ⊆ m1 → ([∗ map] k ↦ x ∈ m1, Φ k x) ⊢ [∗ map] k ↦ x ∈ m2, Φ k x.
  Proof. intros. by apply big_sepL_submseteq, map_to_list_submseteq. Qed.

@@ -339,7 +339,7 @@ Section gmap.

  Lemma big_sepM_empty Φ : ([∗ map] k↦x ∈ ∅, Φ k x) ⊣⊢ emp.
  Proof. by rewrite big_opM_empty. Qed.
-  Lemma big_sepM_empty' `{AffineBI PROP} P Φ : P ⊢ [∗ map] k↦x ∈ ∅, Φ k x.
+  Lemma big_sepM_empty' `{BiAffine PROP} P Φ : P ⊢ [∗ map] k↦x ∈ ∅, Φ k x.
  Proof. rewrite big_sepM_empty. apply: affine. Qed.

  Lemma big_sepM_insert Φ m i x :
@@ -420,16 +420,16 @@ Section gmap.
    ⊢ ([∗ map] k↦x ∈ m, Φ k x) ∧ ([∗ map] k↦x ∈ m, Ψ k x).
  Proof. auto using and_intro, big_sepM_mono, and_elim_l, and_elim_r. Qed.

-  Lemma big_sepM_plainly `{AffineBI PROP} Φ m :
+  Lemma big_sepM_plainly `{BiAffine PROP} Φ m :
    bi_plainly ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ [∗ map] k↦x ∈ m, bi_plainly (Φ k x).
  Proof. apply (big_opM_commute _). Qed.

-  Lemma big_sepM_persistently `{AffineBI PROP} Φ m :
+  Lemma big_sepM_persistently `{BiAffine PROP} Φ m :
    (bi_persistently ([∗ map] k↦x ∈ m, Φ k x)) ⊣⊢
      ([∗ map] k↦x ∈ m, bi_persistently (Φ k x)).
  Proof. apply (big_opM_commute _). Qed.

-  Lemma big_sepM_forall `{AffineBI PROP} Φ m :
+  Lemma big_sepM_forall `{BiAffine PROP} Φ m :
    (∀ k x, Persistent (Φ k x)) →
    ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ (∀ k x, ⌜m !! k = Some x⌝ → Φ k x).
  Proof.
@@ -499,7 +499,7 @@ Section gset.
    (∀ x, x ∈ X → Φ x ⊣⊢ Ψ x) →
    ([∗ set] x ∈ X, Φ x) ⊣⊢ ([∗ set] x ∈ X, Ψ x).
  Proof. apply big_opS_proper. Qed.
-  Lemma big_sepS_subseteq `{AffineBI PROP} Φ X Y :
+  Lemma big_sepS_subseteq `{BiAffine PROP} Φ X Y :
    Y ⊆ X → ([∗ set] x ∈ X, Φ x) ⊢ [∗ set] x ∈ Y, Φ x.
  Proof. intros. by apply big_sepL_submseteq, elements_submseteq. Qed.

@@ -509,7 +509,7 @@ Section gset.

  Lemma big_sepS_empty Φ : ([∗ set] x ∈ ∅, Φ x) ⊣⊢ emp.
  Proof. by rewrite big_opS_empty. Qed.
-  Lemma big_sepS_empty' `{!AffineBI PROP} P Φ : P ⊢ [∗ set] x ∈ ∅, Φ x.
+  Lemma big_sepS_empty' `{!BiAffine PROP} P Φ : P ⊢ [∗ set] x ∈ ∅, Φ x.
  Proof. rewrite big_sepS_empty. apply: affine. Qed.

  Lemma big_sepS_insert Φ X x :
@@ -575,12 +575,12 @@ Section gset.
    by apply sep_mono_r, wand_intro_l.
  Qed.

-  Lemma big_sepS_filter `{AffineBI PROP}
+  Lemma big_sepS_filter `{BiAffine PROP}
      (P : A → Prop) `{∀ x, Decision (P x)} Φ X :
    ([∗ set] y ∈ filter P X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ⌜P y⌝ → Φ y).
  Proof. setoid_rewrite <-decide_emp. apply big_sepS_filter'. Qed.

-  Lemma big_sepS_filter_acc `{AffineBI PROP}
+  Lemma big_sepS_filter_acc `{BiAffine PROP}
      (P : A → Prop) `{∀ y, Decision (P y)} Φ X Y :
    (∀ y, y ∈ Y → P y → y ∈ X) →
    ([∗ set] y ∈ X, Φ y) -∗
@@ -596,15 +596,15 @@ Section gset.
    ([∗ set] y ∈ X, Φ y ∧ Ψ y) ⊢ ([∗ set] y ∈ X, Φ y) ∧ ([∗ set] y ∈ X, Ψ y).
  Proof. auto using and_intro, big_sepS_mono, and_elim_l, and_elim_r. Qed.

-  Lemma big_sepS_plainly `{AffineBI PROP} Φ X :
+  Lemma big_sepS_plainly `{BiAffine PROP} Φ X :
    bi_plainly ([∗ set] y ∈ X, Φ y) ⊣⊢ [∗ set] y ∈ X, bi_plainly (Φ y).
  Proof. apply (big_opS_commute _). Qed.

-  Lemma big_sepS_persistently `{AffineBI PROP} Φ X :
+  Lemma big_sepS_persistently `{BiAffine PROP} Φ X :
    bi_persistently ([∗ set] y ∈ X, Φ y) ⊣⊢ [∗ set] y ∈ X, bi_persistently (Φ y).
  Proof. apply (big_opS_commute _). Qed.

-  Lemma big_sepS_forall `{AffineBI PROP} Φ X :
+  Lemma big_sepS_forall `{BiAffine PROP} Φ X :
    (∀ x, Persistent (Φ x)) → ([∗ set] x ∈ X, Φ x) ⊣⊢ (∀ x, ⌜x ∈ X⌝ → Φ x).
  Proof.
    intros. apply (anti_symm _).
@@ -671,7 +671,7 @@ Section gmultiset.
    (∀ x, x ∈ X → Φ x ⊣⊢ Ψ x) →
    ([∗ mset] x ∈ X, Φ x) ⊣⊢ ([∗ mset] x ∈ X, Ψ x).
  Proof. apply big_opMS_proper. Qed.
-  Lemma big_sepMS_subseteq `{AffineBI PROP} Φ X Y :
+  Lemma big_sepMS_subseteq `{BiAffine PROP} Φ X Y :
    Y ⊆ X → ([∗ mset] x ∈ X, Φ x) ⊢ [∗ mset] x ∈ Y, Φ x.
  Proof. intros. by apply big_sepL_submseteq, gmultiset_elements_submseteq. Qed.

@@ -681,7 +681,7 @@ Section gmultiset.

  Lemma big_sepMS_empty Φ : ([∗ mset] x ∈ ∅, Φ x) ⊣⊢ emp.
  Proof. by rewrite big_opMS_empty. Qed.
-  Lemma big_sepMS_empty' `{!AffineBI PROP} P Φ : P ⊢ [∗ mset] x ∈ ∅, Φ x.
+  Lemma big_sepMS_empty' `{!BiAffine PROP} P Φ : P ⊢ [∗ mset] x ∈ ∅, Φ x.
  Proof. rewrite big_sepMS_empty. apply: affine. Qed.

  Lemma big_sepMS_union Φ X Y :
@@ -714,11 +714,11 @@ Section gmultiset.
    ([∗ mset] y ∈ X, Φ y ∧ Ψ y) ⊢ ([∗ mset] y ∈ X, Φ y) ∧ ([∗ mset] y ∈ X, Ψ y).
  Proof. auto using and_intro, big_sepMS_mono, and_elim_l, and_elim_r. Qed.

-  Lemma big_sepMS_plainly `{AffineBI PROP} Φ X :
+  Lemma big_sepMS_plainly `{BiAffine PROP} Φ X :
    bi_plainly ([∗ mset] y ∈ X, Φ y) ⊣⊢ [∗ mset] y ∈ X, bi_plainly (Φ y).
  Proof. apply (big_opMS_commute _). Qed.

-  Lemma big_sepMS_persistently `{AffineBI PROP} Φ X :
+  Lemma big_sepMS_persistently `{BiAffine PROP} Φ X :
    bi_persistently ([∗ mset] y ∈ X, Φ y) ⊣⊢
      [∗ mset] y ∈ X, bi_persistently (Φ y).
  Proof. apply (big_opMS_commute _). Qed.
@@ -756,11 +756,11 @@ Section list.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat → A → PROP.

-  Lemma big_sepL_later `{AffineBI PROP} Φ l :
+  Lemma big_sepL_later `{BiAffine PROP} Φ l :
    ▷ ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, ▷ Φ k x).
  Proof. apply (big_opL_commute _). Qed.

-  Lemma big_sepL_laterN `{AffineBI PROP} Φ n l :
+  Lemma big_sepL_laterN `{BiAffine PROP} Φ n l :
    ▷^n ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, ▷^n Φ k x).
  Proof. apply (big_opL_commute _). Qed.

@@ -781,11 +781,11 @@ Section gmap.
  Implicit Types m : gmap K A.
  Implicit Types Φ Ψ : K → A → PROP.

-  Lemma big_sepM_later `{AffineBI PROP} Φ m :
+  Lemma big_sepM_later `{BiAffine PROP} Φ m :
    ▷ ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ ([∗ map] k↦x ∈ m, ▷ Φ k x).
  Proof. apply (big_opM_commute _). Qed.

-  Lemma big_sepM_laterN `{AffineBI PROP} Φ n m :
+  Lemma big_sepM_laterN `{BiAffine PROP} Φ n m :
    ▷^n ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ ([∗ map] k↦x ∈ m, ▷^n Φ k x).
  Proof. apply (big_opM_commute _). Qed.

@@ -803,11 +803,11 @@ Section gset.
  Implicit Types X : gset A.
  Implicit Types Φ : A → PROP.

-  Lemma big_sepS_later `{AffineBI PROP} Φ X :
+  Lemma big_sepS_later `{BiAffine PROP} Φ X :
    ▷ ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ▷ Φ y).
  Proof. apply (big_opS_commute _). Qed.

-  Lemma big_sepS_laterN `{AffineBI PROP} Φ n X :
+  Lemma big_sepS_laterN `{BiAffine PROP} Φ n X :
    ▷^n ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ▷^n Φ y).
  Proof. apply (big_opS_commute _). Qed.

@@ -825,11 +825,11 @@ Section gmultiset.
  Implicit Types X : gmultiset A.
  Implicit Types Φ : A → PROP.

-  Lemma big_sepMS_later `{AffineBI PROP} Φ X :
+  Lemma big_sepMS_later `{BiAffine PROP} Φ X :
    ▷ ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, ▷ Φ y).
  Proof. apply (big_opMS_commute _). Qed.

-  Lemma big_sepMS_laterN `{AffineBI PROP} Φ n X :
+  Lemma big_sepMS_laterN `{BiAffine PROP} Φ n X :
    ▷^n ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, ▷^n Φ y).
  Proof. apply (big_opMS_commute _). Qed.


--- a/theories/bi/counter_examples.v
+++ b/theories/bi/counter_examples.v
@@ -5,7 +5,7 @@ Set Default Proof Using "Type*".
 (** This proves that we need the ▷ in a "Saved Proposition" construction with
 name-dependent allocation. *)
 Module savedprop. Section savedprop.
-  Context `{AffineBI PROP}.
+  Context `{BiAffine PROP}.
  Notation "¬ P" := (□ (P → False))%I : bi_scope.
  Implicit Types P : PROP.

@@ -65,7 +65,7 @@ End savedprop. End savedprop.

 (** This proves that we need the ▷ when opening invariants. *)
 Module inv. Section inv.
-  Context `{AffineBI PROP}.
+  Context `{BiAffine PROP}.
  Implicit Types P : PROP.

  (** Assumptions *)

--- a/theories/bi/derived.v
+++ b/theories/bi/derived.v
@@ -39,11 +39,11 @@ Arguments Affine {_} _%I : simpl never.
 Arguments affine {_} _%I {_}.
 Hint Mode Affine + ! : typeclass_instances.

-Class AffineBI (PROP : bi) := absorbing_bi (Q : PROP) : Affine Q.
+Class BiAffine (PROP : bi) := absorbing_bi (Q : PROP) : Affine Q.
 Existing Instance absorbing_bi | 0.

-Class PositiveBI (PROP : bi) :=
-  positive_bi (P Q : PROP) : bi_affinely (P ∗ Q) ⊢ bi_affinely P ∗ Q.
+Class BiPositive (PROP : bi) :=
+  bi_positive (P Q : PROP) : bi_affinely (P ∗ Q) ⊢ bi_affinely P ∗ Q.

 Definition bi_absorbingly {PROP : bi} (P : PROP) : PROP := (True ∗ P)%I.
 Arguments bi_absorbingly {_} _%I : simpl never.
@@ -732,11 +732,11 @@ Proof.
  - by rewrite !and_elim_l right_id.
  - by rewrite !and_elim_r.
 Qed.
-Lemma affinely_sep `{PositiveBI PROP} P Q :
+Lemma affinely_sep `{BiPositive PROP} P Q :
  bi_affinely (P ∗ Q) ⊣⊢ bi_affinely P ∗ bi_affinely Q.
 Proof.
  apply (anti_symm _), affinely_sep_2.
-  by rewrite -{1}affinely_idemp positive_bi !(comm _ (bi_affinely P)%I) positive_bi.
+  by rewrite -{1}affinely_idemp bi_positive !(comm _ (bi_affinely P)%I) bi_positive.
 Qed.
 Lemma affinely_forall {A} (Φ : A → PROP) :
  bi_affinely (∀ a, Φ a) ⊢ ∀ a, bi_affinely (Φ a).
@@ -815,7 +815,7 @@ Proof. by rewrite /bi_absorbingly !assoc (comm _ P). Qed.
 Lemma absorbingly_sep_lr P Q : bi_absorbingly P ∗ Q ⊣⊢ P ∗ bi_absorbingly Q.
 Proof. by rewrite absorbingly_sep_l absorbingly_sep_r. Qed.

-Lemma affinely_absorbingly `{!PositiveBI PROP} P :
+Lemma affinely_absorbingly `{!BiPositive PROP} P :
  bi_affinely (bi_absorbingly P) ⊣⊢ bi_affinely P.
 Proof.
  apply (anti_symm _), affinely_mono, absorbingly_intro.
@@ -875,12 +875,12 @@ Proof. apply (anti_symm _); auto using True_sep_2. Qed.
 Lemma sep_True P `{!Absorbing P} : P ∗ True ⊣⊢ P.
 Proof. by rewrite comm True_sep. Qed.

-Section affine_bi.
-  Context `{AffineBI PROP}.
+Section bi_affine.
+  Context `{BiAffine PROP}.

-  Global Instance affine_bi_absorbing P : Absorbing P | 0.
+  Global Instance bi_affine_absorbing P : Absorbing P | 0.
  Proof. by rewrite /Absorbing /bi_absorbingly (affine True%I) left_id. Qed.
-  Global Instance affine_bi_positive : PositiveBI PROP.
+  Global Instance bi_affine_positive : BiPositive PROP.
  Proof. intros P Q. by rewrite !affine_affinely. Qed.

  Lemma True_emp : True ⊣⊢ emp.
@@ -906,7 +906,7 @@ Section affine_bi.
    - by rewrite pure_True // True_impl.
    - by rewrite pure_False // False_impl True_emp.
  Qed.
-End affine_bi.
+End bi_affine.

 (* Properties of the persistence modality *)
 Hint Resolve persistently_mono.
@@ -1044,7 +1044,7 @@ Qed.
 Lemma persistently_sep_2 P Q :
  bi_persistently P ∗ bi_persistently Q ⊢ bi_persistently (P ∗ Q).
 Proof. by rewrite -persistently_and_sep persistently_and -and_sep_persistently. Qed.
-Lemma persistently_sep `{PositiveBI PROP} P Q :
+Lemma persistently_sep `{BiPositive PROP} P Q :
  bi_persistently (P ∗ Q) ⊣⊢ bi_persistently P ∗ bi_persistently Q.
 Proof.
  apply (anti_symm _); auto using persistently_sep_2.
@@ -1076,7 +1076,7 @@ Lemma impl_wand_persistently_2 P Q : (bi_persistently P -∗ Q) ⊢ (bi_persiste
 Proof. apply impl_intro_l. by rewrite persistently_and_sep_l_1 wand_elim_r. Qed.

 Section persistently_affinely_bi.
-  Context `{AffineBI PROP}.
+  Context `{BiAffine PROP}.

  Lemma persistently_emp : bi_persistently emp ⊣⊢ emp.
  Proof. by rewrite -!True_emp persistently_pure. Qed.
@@ -1243,7 +1243,7 @@ Qed.
 Lemma plainly_sep_2 P Q :
  bi_plainly P ∗ bi_plainly Q ⊢ bi_plainly (P ∗ Q).
 Proof. by rewrite -plainly_and_sep plainly_and -and_sep_plainly. Qed.
-Lemma plainly_sep `{PositiveBI PROP} P Q :
+Lemma plainly_sep `{BiPositive PROP} P Q :
  bi_plainly (P ∗ Q) ⊣⊢ bi_plainly P ∗ bi_plainly Q.
 Proof.
  apply (anti_symm _); auto using plainly_sep_2.
@@ -1271,7 +1271,7 @@ Lemma impl_wand_plainly_2 P Q : (bi_plainly P -∗ Q) ⊢ (bi_plainly P → Q).
 Proof. apply impl_intro_l. by rewrite plainly_and_sep_l_1 wand_elim_r. Qed.

 Section plainly_affinely_bi.
-  Context `{AffineBI PROP}.
+  Context `{BiAffine PROP}.

  Lemma plainly_emp : bi_plainly emp ⊣⊢ emp.
  Proof. by rewrite -!True_emp plainly_pure. Qed.
@@ -1316,7 +1316,7 @@ Lemma affinely_persistently_exist {A} (Φ : A → PROP) : □ (∃ x, Φ x) ⊣
 Proof. by rewrite persistently_exist affinely_exist. Qed.
 Lemma affinely_persistently_sep_2 P Q : □ P ∗ □ Q ⊢ □ (P ∗ Q).
 Proof. by rewrite affinely_sep_2 persistently_sep_2. Qed.
-Lemma affinely_persistently_sep `{PositiveBI PROP} P Q : □ (P ∗ Q) ⊣⊢ □ P ∗ □ Q.
+Lemma affinely_persistently_sep `{BiPositive PROP} P Q : □ (P ∗ Q) ⊣⊢ □ P ∗ □ Q.
 Proof. by rewrite -affinely_sep -persistently_sep. Qed.

 Lemma affinely_persistently_idemp P : □ □ P ⊣⊢ □ P.
@@ -1368,7 +1368,7 @@ Lemma affinely_plainly_exist {A} (Φ : A → PROP) : ■ (∃ x, Φ x) ⊣⊢ 
 Proof. by rewrite plainly_exist affinely_exist. Qed.
 Lemma affinely_plainly_sep_2 P Q : ■ P ∗ ■ Q ⊢ ■ (P ∗ Q).
 Proof. by rewrite affinely_sep_2 plainly_sep_2. Qed.
-Lemma affinely_plainly_sep `{PositiveBI PROP} P Q : ■ (P ∗ Q) ⊣⊢ ■ P ∗ ■ Q.
+Lemma affinely_plainly_sep `{BiPositive PROP} P Q : ■ (P ∗ Q) ⊣⊢ ■ P ∗ ■ Q.
 Proof. by rewrite -affinely_sep -plainly_sep. Qed.

 Lemma affinely_plainly_idemp P : ■ ■ P ⊣⊢ ■ P.
@@ -1427,7 +1427,7 @@ Proof. destruct p; simpl; auto using affinely_exist. Qed.
 Lemma affinely_if_sep_2 p P Q :
  bi_affinely_if p P ∗ bi_affinely_if p Q ⊢ bi_affinely_if p (P ∗ Q).
 Proof. destruct p; simpl; auto using affinely_sep_2. Qed.
-Lemma affinely_if_sep `{PositiveBI PROP} p P Q :
+Lemma affinely_if_sep `{BiPositive PROP} p P Q :
  bi_affinely_if p (P ∗ Q) ⊣⊢ bi_affinely_if p P ∗ bi_affinely_if p Q.
 Proof. destruct p; simpl; auto using affinely_sep. Qed.

@@ -1466,7 +1466,7 @@ Proof. destruct p; simpl; auto using persistently_exist. Qed.
 Lemma persistently_if_sep_2 p P Q :
  bi_persistently_if p P ∗ bi_persistently_if p Q ⊢ bi_persistently_if p (P ∗ Q).
 Proof. destruct p; simpl; auto using persistently_sep_2. Qed.
-Lemma persistently_if_sep `{PositiveBI PROP} p P Q :
+Lemma persistently_if_sep `{BiPositive PROP} p P Q :
  bi_persistently_if p (P ∗ Q) ⊣⊢ bi_persistently_if p P ∗ bi_persistently_if p Q.
 Proof. destruct p; simpl; auto using persistently_sep. Qed.

@@ -1496,7 +1496,7 @@ Lemma affinely_persistently_if_exist {A} p (Ψ : A → PROP) :
 Proof. destruct p; simpl; auto using affinely_persistently_exist. Qed.
 Lemma affinely_persistently_if_sep_2 p P Q : □?p P ∗ □?p Q ⊢ □?p (P ∗ Q).
 Proof. destruct p; simpl; auto using affinely_persistently_sep_2. Qed.
-Lemma affinely_persistently_if_sep `{PositiveBI PROP} p P Q :
+Lemma affinely_persistently_if_sep `{BiPositive PROP} p P Q :
  □?p (P ∗ Q) ⊣⊢ □?p P ∗ □?p Q.
 Proof. destruct p; simpl; auto using affinely_persistently_sep. Qed.

@@ -1527,7 +1527,7 @@ Lemma plainly_if_or p P Q : bi_plainly_if p (P ∨ Q) ⊣⊢ bi_plainly_if p P 
 Proof. destruct p; simpl; auto using plainly_or. Qed.
 Lemma plainly_if_exist {A} p (Ψ : A → PROP) : (bi_plainly_if p (∃ a, Ψ a)) ⊣⊢ ∃ a, bi_plainly_if p (Ψ a).
 Proof. destruct p; simpl; auto using plainly_exist. Qed.
-Lemma plainly_if_sep `{PositiveBI PROP} p P Q :
+Lemma plainly_if_sep `{BiPositive PROP} p P Q :
  bi_plainly_if p (P ∗ Q) ⊣⊢ bi_plainly_if p P ∗ bi_plainly_if p Q.
 Proof. destruct p; simpl; auto using plainly_sep. Qed.

@@ -1557,7 +1557,7 @@ Lemma affinely_plainly_if_exist {A} p (Ψ : A → PROP) :
 Proof. destruct p; simpl; auto using affinely_plainly_exist. Qed.
 Lemma affinely_plainly_if_sep_2 p P Q : ■?p P ∗ ■?p Q ⊢ ■?p (P ∗ Q).
 Proof. destruct p; simpl; auto using affinely_plainly_sep_2. Qed.
-Lemma affinely_plainly_if_sep `{PositiveBI PROP} p P Q :
+Lemma affinely_plainly_if_sep `{BiPositive PROP} p P Q :
  ■?p (P ∗ Q) ⊣⊢ ■?p P ∗ ■?p Q.
 Proof. destruct p; simpl; auto using affinely_plainly_sep. Qed.

@@ -1626,7 +1626,7 @@ Lemma impl_wand_2 P `{!Persistent P} Q : (P -∗ Q) ⊢ P → Q.
 Proof. apply impl_intro_l. by rewrite persistent_and_sep_1 wand_elim_r. Qed.

 Section persistent_bi_absorbing.
-  Context `{AffineBI PROP}.
+  Context `{BiAffine PROP}.

  Lemma persistent_and_sep P Q `{HPQ : !TCOr (Persistent P) (Persistent Q)} :
    P ∧ Q ⊣⊢ P ∗ Q.
@@ -1858,11 +1858,11 @@ Proof.
  split; [split|]; try apply _. apply plainly_or. apply plainly_pure.
 Qed.

-Global Instance bi_plainly_sep_weak_homomorphism `{PositiveBI PROP} :
+Global Instance bi_plainly_sep_weak_homomorphism `{BiPositive PROP} :
  WeakMonoidHomomorphism bi_sep bi_sep (≡) (@bi_plainly PROP).
 Proof. split; try apply _. apply plainly_sep. Qed.

-Global Instance bi_plainly_sep_homomorphism `{AffineBI PROP} :
+Global Instance bi_plainly_sep_homomorphism `{BiAffine PROP} :
  MonoidHomomorphism bi_sep bi_sep (≡) (@bi_plainly PROP).
 Proof. split. apply _. apply plainly_emp. Qed.

@@ -1886,11 +1886,11 @@ Proof.
  split; [split|]; try apply _. apply persistently_or. apply persistently_pure.
 Qed.

-Global Instance bi_persistently_sep_weak_homomorphism `{PositiveBI PROP} :
+Global Instance bi_persistently_sep_weak_homomorphism `{BiPositive PROP} :
  WeakMonoidHomomorphism bi_sep bi_sep (≡) (@bi_persistently PROP).
 Proof. split; try apply _. apply persistently_sep. Qed.

-Global Instance bi_persistently_sep_homomorphism `{AffineBI PROP} :
+Global Instance bi_persistently_sep_homomorphism `{BiAffine PROP} :
  MonoidHomomorphism bi_sep bi_sep (≡) (@bi_persistently PROP).
 Proof. split. apply _. apply persistently_emp. Qed.

@@ -1978,7 +1978,7 @@ Qed.

 Lemma later_True : ▷ True ⊣⊢ True.
 Proof. apply (anti_symm (⊢)); auto using later_intro. Qed.
-Lemma later_emp `{!AffineBI PROP} : ▷ emp ⊣⊢ emp.
+Lemma later_emp `{!BiAffine PROP} : ▷ emp ⊣⊢ emp.
 Proof. by rewrite -True_emp later_True. Qed.
 Lemma later_forall {A} (Φ : A → PROP) : (▷ ∀ a, Φ a) ⊣⊢ (∀ a, ▷ Φ a).
 Proof.
@@ -2059,7 +2059,7 @@ Proof. induction n as [|n IH]; simpl; by rewrite -?later_intro. Qed.

 Lemma laterN_True n : ▷^n True ⊣⊢ True.
 Proof. apply (anti_symm (⊢)); auto using laterN_intro, True_intro. Qed.
-Lemma laterN_emp `{!AffineBI PROP} n : ▷^n emp ⊣⊢ emp.
+Lemma laterN_emp `{!BiAffine PROP} n : ▷^n emp ⊣⊢ emp.
 Proof. by rewrite -True_emp laterN_True. Qed.
 Lemma laterN_forall {A} n (Φ : A → PROP) : (▷^n ∀ a, Φ a) ⊣⊢ (∀ a, ▷^n Φ a).
 Proof. induction n as [|n IH]; simpl; rewrite -?later_forall ?IH; auto. Qed.
@@ -2124,7 +2124,7 @@ Proof. apply (anti_symm _); rewrite /bi_except_0; auto. Qed.

 Lemma except_0_True : ◇ True ⊣⊢ True.
 Proof. rewrite /bi_except_0. apply (anti_symm _); auto. Qed.
-Lemma except_0_emp `{!AffineBI PROP} : ◇ emp ⊣⊢ emp.
+Lemma except_0_emp `{!BiAffine PROP} : ◇ emp ⊣⊢ emp.
 Proof. by rewrite -True_emp except_0_True. Qed.
 Lemma except_0_or P Q : ◇ (P ∨ Q) ⊣⊢ ◇ P ∨ ◇ Q.
 Proof. rewrite /bi_except_0. apply (anti_symm _); auto. Qed.
@@ -2208,7 +2208,7 @@ Proof.
  rewrite /Timeless /bi_except_0 pure_alt later_exist_false.
  apply or_elim, exist_elim; [auto|]=> Hφ. rewrite -(exist_intro Hφ). auto.
 Qed.
-Global Instance emp_timeless `{AffineBI PROP} : Timeless (emp : PROP)%I.
+Global Instance emp_timeless `{BiAffine PROP} : Timeless (emp : PROP)%I.
 Proof. rewrite -True_emp. apply _. Qed.

 Global Instance and_timeless P Q : Timeless P → Timeless Q → Timeless (P ∧ Q).
@@ -2305,13 +2305,13 @@ Global Instance bi_except_0_sep_weak_homomorphism :
  WeakMonoidHomomorphism bi_sep bi_sep (≡) (@bi_except_0 PROP).
 Proof. split; try apply _. apply except_0_sep. Qed.

-Global Instance bi_later_monoid_sep_homomorphism `{!AffineBI PROP} :
+Global Instance bi_later_monoid_sep_homomorphism `{!BiAffine PROP} :
  MonoidHomomorphism bi_sep bi_sep (≡) (@bi_later PROP).
 Proof. split; try apply _. apply later_emp. Qed.
-Global Instance bi_laterN_sep_homomorphism `{!AffineBI PROP} n :
+Global Instance bi_laterN_sep_homomorphism `{!BiAffine PROP} n :
  MonoidHomomorphism bi_sep bi_sep (≡) (@bi_laterN PROP n).
 Proof. split; try apply _. apply laterN_emp. Qed.
-Global Instance bi_except_0_sep_homomorphism `{!AffineBI PROP} :
+Global Instance bi_except_0_sep_homomorphism `{!BiAffine PROP} :
  MonoidHomomorphism bi_sep bi_sep (≡) (@bi_except_0 PROP).
 Proof. split; try apply _. apply except_0_emp. Qed.


--- a/theories/bi/tactics.v
+++ b/theories/bi/tactics.v
@@ -32,7 +32,7 @@ Module bi_reflection. Section bi_reflection.
  Qed.

  (* Can be related to the RHS being affine *)
-  Lemma flatten_entails `{AffineBI PROP} Σ e1 e2 :
+  Lemma flatten_entails `{BiAffine PROP} Σ e1 e2 :
    flatten e2 ⊆+ flatten e1 → eval Σ e1 ⊢ eval Σ e2.
  Proof. intros. rewrite !eval_flatten. by apply big_sepL_submseteq. Qed.
  Lemma flatten_equiv Σ e1 e2 :

--- a/theories/proofmode/class_instances.v
+++ b/theories/proofmode/class_instances.v
@@ -50,7 +50,7 @@ Proof.
  rewrite /FromAssumption /= =><-.
  by rewrite persistently_elim plainly_elim_persistently.
 Qed.
-Global Instance from_assumption_plainly_l_false `{AffineBI PROP} P Q :
+Global Instance from_assumption_plainly_l_false `{BiAffine PROP} P Q :
  FromAssumption true P Q → FromAssumption false (bi_plainly P) Q.
 Proof.
  rewrite /FromAssumption /= =><-.
@@ -62,7 +62,7 @@ Proof. rewrite /FromAssumption /= =><-. by rewrite affinely_persistently_if_elim
 Global Instance from_assumption_persistently_l_true P Q :
  FromAssumption true P Q → FromAssumption true (bi_persistently P) Q.
 Proof. rewrite /FromAssumption /= =><-. by rewrite persistently_idemp. Qed.
-Global Instance from_assumption_persistently_l_false `{AffineBI PROP} P Q :
+Global Instance from_assumption_persistently_l_false `{BiAffine PROP} P Q :
  FromAssumption true P Q → FromAssumption false (bi_persistently P) Q.
 Proof. rewrite /FromAssumption /= =><-. by rewrite affine_affinely. Qed.
 Global Instance from_assumption_affinely_l_true p P Q :
@@ -224,7 +224,7 @@ Proof.
  rewrite /FromAssumption /IntoWand=> HP. by rewrite HP affinely_persistently_if_elim.
 Qed.

-Global Instance into_wand_impl_false_false `{!AffineBI PROP} P Q P' :
+Global Instance into_wand_impl_false_false `{!BiAffine PROP} P Q P' :
  FromAssumption false P P' → IntoWand false false (P' → Q) P Q.