Force all the BI morphisms to use the ⎡P⎤ notation.

theories/bi/derived_laws.v

@@ -2299,38 +2299,38 @@ Hint Immediate bi.plain_persistent : typeclass_instances.
 
 (* BI morphisms *)
 Section bi_morphims.
-  Context `{@BiMorphism PROP1 PROP2 mor}.
+  Context `{BiMorphism PROP1 PROP2}.
 
-  Global Instance bi_mor_proper : Proper ((≡) ==> (≡)) mor.
+  Global Instance bi_mor_proper : Proper ((≡) ==> (≡)) bi_embedding.
   Proof. apply (ne_proper _). Qed.
-  Global Instance bi_mor_mono_flip : Proper (flip (⊢) ==> flip (⊢)) mor.
+  Global Instance bi_mor_mono_flip : Proper (flip (⊢) ==> flip (⊢)) bi_embedding.
   Proof. solve_proper. Qed.
 
-  Lemma bi_mor_forall A (Φ : A → PROP1) : mor (∀ x, Φ x) ⊣⊢ (∀ x, mor (Φ x)).
+  Lemma bi_mor_forall A (Φ : A → PROP1) : ⎡∀ x, Φ x⎤ ⊣⊢ ∀ x, ⎡Φ x⎤.
   Proof.
     apply bi.equiv_spec; split; [|apply bi_mor_forall_2].
     apply bi.forall_intro=>?. by rewrite bi.forall_elim.
   Qed.
-  Lemma bi_mor_exist A (Φ : A → PROP1) : mor (∃ x, Φ x) ⊣⊢ (∃ x, mor (Φ x)).
+  Lemma bi_mor_exist A (Φ : A → PROP1) : ⎡∃ x, Φ x⎤ ⊣⊢ ∃ x, ⎡Φ x⎤.
   Proof.
     apply bi.equiv_spec; split; [apply bi_mor_exist_1|].
     apply bi.exist_elim=>?. by rewrite -bi.exist_intro.
   Qed.
-  Lemma bi_mor_and P Q : mor (P ∧ Q) ⊣⊢ mor P ∧ mor Q.
+  Lemma bi_mor_and P Q : ⎡P ∧ Q⎤ ⊣⊢ ⎡P⎤ ∧ ⎡Q⎤.
   Proof. rewrite !bi.and_alt bi_mor_forall. by f_equiv=>-[]. Qed.
-  Lemma bi_mor_or P Q : mor (P ∨ Q) ⊣⊢ mor P ∨ mor Q.
+  Lemma bi_mor_or P Q : ⎡P ∨ Q⎤ ⊣⊢ ⎡P⎤ ∨ ⎡Q⎤.
   Proof. rewrite !bi.or_alt bi_mor_exist. by f_equiv=>-[]. Qed.
-  Lemma bi_mor_impl P Q : mor (P → Q) ⊣⊢ (mor P → mor Q).
+  Lemma bi_mor_impl P Q : ⎡P → Q⎤ ⊣⊢ (⎡P⎤ → ⎡Q⎤).
   Proof.
     apply bi.equiv_spec; split; [|apply bi_mor_impl_2].
     apply bi.impl_intro_l. by rewrite -bi_mor_and bi.impl_elim_r.
   Qed.
-  Lemma bi_mor_wand P Q : mor (P -∗ Q) ⊣⊢ (mor P -∗ mor Q).
+  Lemma bi_mor_wand P Q : ⎡P -∗ Q⎤ ⊣⊢ (⎡P⎤ -∗ ⎡Q⎤).
   Proof.
     apply bi.equiv_spec; split; [|apply bi_mor_wand_2].
     apply bi.wand_intro_l. by rewrite -bi_mor_sep bi.wand_elim_r.
   Qed.
-  Lemma bi_mor_pure φ : mor ⌜φ⌝ ⊣⊢ ⌜φ⌝.
+  Lemma bi_mor_pure φ : ⎡⌜φ⌝⎤ ⊣⊢ ⌜φ⌝.
   Proof.
     rewrite (@bi.pure_alt PROP1) (@bi.pure_alt PROP2) bi_mor_exist.
     do 2 f_equiv. apply bi.equiv_spec. split; [apply bi.True_intro|].
@@ -2338,56 +2338,54 @@ Section bi_morphims.
       last apply bi.True_intro.
     apply bi.impl_intro_l. by rewrite right_id.
   Qed.
-  Lemma bi_mor_internal_eq (A : ofeT) (x y : A) : mor (x ≡ y) ⊣⊢ (x ≡ y).
+  Lemma bi_mor_internal_eq (A : ofeT) (x y : A) : ⎡x ≡ y⎤ ⊣⊢ x ≡ y.
   Proof.
     apply bi.equiv_spec; split; [apply bi_mor_internal_eq_1|].
-    rewrite (bi.internal_eq_rewrite x y (λ y, mor (x ≡ y)%I) _); [|solve_proper].
+    etrans; [apply (bi.internal_eq_rewrite x y (λ y, ⎡x ≡ y⎤%I)); solve_proper|].
     rewrite -(bi.internal_eq_refl True%I) bi_mor_pure.
     eapply bi.impl_elim; [done|]. apply bi.True_intro.
   Qed.
-  Lemma bi_mor_iff P Q : mor (P ↔ Q) ⊣⊢ (mor P ↔ mor Q).
+  Lemma bi_mor_iff P Q : ⎡P ↔ Q⎤ ⊣⊢ (⎡P⎤ ↔ ⎡Q⎤).
   Proof. by rewrite bi_mor_and !bi_mor_impl. Qed.
-  Lemma bi_mor_wand_iff P Q : mor (P ∗-∗ Q) ⊣⊢ (mor P ∗-∗ mor Q).
+  Lemma bi_mor_wand_iff P Q : ⎡P ∗-∗ Q⎤ ⊣⊢ (⎡P⎤ ∗-∗ ⎡Q⎤).
   Proof. by rewrite bi_mor_and !bi_mor_wand. Qed.
-  Lemma bi_mor_affinely P : mor (bi_affinely P) ⊣⊢ bi_affinely (mor P).
+  Lemma bi_mor_affinely P : ⎡bi_affinely P⎤ ⊣⊢ bi_affinely ⎡P⎤.
   Proof. by rewrite bi_mor_and bi_mor_emp. Qed.
-  Lemma bi_mor_absorbingly P : mor (bi_absorbingly P) ⊣⊢ bi_absorbingly (mor P).
+  Lemma bi_mor_absorbingly P : ⎡bi_absorbingly P⎤ ⊣⊢ bi_absorbingly ⎡P⎤.
   Proof. by rewrite bi_mor_sep bi_mor_pure. Qed.
-  Lemma bi_mor_plainly_if P b :
-    mor (bi_plainly_if b P) ⊣⊢ bi_plainly_if b (mor P).
+  Lemma bi_mor_plainly_if P b : ⎡bi_plainly_if b P⎤ ⊣⊢ bi_plainly_if b ⎡P⎤.
   Proof. destruct b; auto using bi_mor_plainly. Qed.
   Lemma bi_mor_persistently_if P b :
-    mor (bi_persistently_if b P) ⊣⊢ bi_persistently_if b (mor P).
+    ⎡bi_persistently_if b P⎤ ⊣⊢ bi_persistently_if b ⎡P⎤.
   Proof. destruct b; auto using bi_mor_persistently. Qed.
-  Lemma bi_mor_affinely_if P b :
-    mor (bi_affinely_if b P) ⊣⊢ bi_affinely_if b (mor P).
+  Lemma bi_mor_affinely_if P b : ⎡bi_affinely_if b P⎤ ⊣⊢ bi_affinely_if b ⎡P⎤.
   Proof. destruct b; simpl; auto using bi_mor_affinely. Qed.
   Lemma bi_mor_hforall {As} (Φ : himpl As PROP1):
-    mor (bi_hforall Φ) ⊣⊢ bi_hforall (hcompose mor Φ).
+    ⎡bi_hforall Φ⎤ ⊣⊢ bi_hforall (hcompose bi_embedding Φ).
   Proof. induction As=>//. rewrite /= bi_mor_forall. by do 2 f_equiv. Qed.
   Lemma bi_mor_hexist {As} (Φ : himpl As PROP1):
-    mor (bi_hexist Φ) ⊣⊢ bi_hexist (hcompose mor Φ).
+    ⎡bi_hexist Φ⎤ ⊣⊢ bi_hexist (hcompose bi_embedding Φ).
   Proof. induction As=>//. rewrite /= bi_mor_exist. by do 2 f_equiv. Qed.
 
-  Global Instance bi_mor_plain P : Plain P → Plain (mor P).
+  Global Instance bi_mor_plain P : Plain P → Plain ⎡P⎤.
   Proof. intros ?. by rewrite /Plain -bi_mor_plainly -plain. Qed.
-  Global Instance bi_mor_persistent P : Persistent P → Persistent (mor P).
+  Global Instance bi_mor_persistent P : Persistent P → Persistent ⎡P⎤.
   Proof. intros ?. by rewrite /Persistent -bi_mor_persistently -persistent. Qed.
-  Global Instance bi_mor_affine P : Affine P → Affine (mor P).
+  Global Instance bi_mor_affine P : Affine P → Affine ⎡P⎤.
   Proof. intros ?. by rewrite /Affine (affine P) bi_mor_emp. Qed.
-  Global Instance bi_mor_absorbing P : Absorbing P → Absorbing (mor P).
+  Global Instance bi_mor_absorbing P : Absorbing P → Absorbing ⎡P⎤.
   Proof. intros ?. by rewrite /Absorbing -bi_mor_absorbingly absorbing. Qed.
 End bi_morphims.
 
 Section sbi_morphims.
-  Context `{@SbiMorphism PROP1 PROP2 mor}.
+  Context `{SbiMorphism PROP1 PROP2}.
 
-  Lemma sbi_mor_laterN n P : mor (▷^n P) ⊣⊢ ▷^n (mor P).
+  Lemma sbi_mor_laterN n P : ⎡▷^n P⎤ ⊣⊢ ▷^n ⎡P⎤.
   Proof. induction n=>//=. rewrite sbi_mor_later. by f_equiv. Qed.
-  Lemma sbi_mor_except_0 P : mor (◇ P) ⊣⊢ ◇ (mor P).
+  Lemma sbi_mor_except_0 P : ⎡◇ P⎤ ⊣⊢ ◇ ⎡P⎤.
   Proof. by rewrite bi_mor_or sbi_mor_later bi_mor_pure. Qed.
 
-  Global Instance sbi_mor_timeless P : Timeless P → Timeless (mor P).
+  Global Instance sbi_mor_timeless P : Timeless P → Timeless ⎡P⎤.
   Proof.
     intros ?. by rewrite /Timeless -sbi_mor_except_0 -sbi_mor_later timeless.
   Qed.

theories/bi/interface.v

@@ -526,28 +526,28 @@ End sbi_laws.
 End bi.
 
 (* Typically, embeddings are used to *define* the destination BI.
-   Hence we cannot ask B to be a BI. *)
+   Hence we cannot ask it to be a morphism. *)
 Class BiEmbedding (A B : Type) := bi_embedding : A → B.
 Arguments bi_embedding {_ _ _} _%I : simpl never.
 Notation "⎡ P ⎤" := (bi_embedding P) : bi_scope.
 Instance: Params (@bi_embedding) 3.
 Typeclasses Opaque bi_embedding.
 
-Class BiMorphism {PROP1 PROP2 : bi} (f : PROP1 → PROP2) := {
-  bi_mor_ne :> NonExpansive f;
-  bi_mor_mono :> Proper ((⊢) ==> (⊢)) f;
-  bi_mor_emp : f emp ⊣⊢ emp;
-  bi_mor_impl_2 P Q : (f P → f Q)%I ⊢ f (P → Q)%I;
-  bi_mor_forall_2 A (Φ : A → PROP1) : (∀ x, f (Φ x)) ⊢ f (∀ x, Φ x);
-  bi_mor_exist_1 A (Φ : A → PROP1) : f (∃ x, Φ x) ⊢ ∃ x, f (Φ x);
-  bi_mor_internal_eq_1 (A : ofeT) (x y : A) : f (x ≡ y) ⊢ (x ≡ y);
-  bi_mor_sep P Q : f (P ∗ Q) ⊣⊢ (f P ∗ f Q);
-  bi_mor_wand_2 P Q : (f P -∗ f Q) ⊢ f (P -∗ Q);
-  bi_mor_plainly P : f (bi_plainly P) ⊣⊢ bi_plainly (f P);
-  bi_mor_persistently P : f (bi_persistently P) ⊣⊢ bi_persistently (f P)
+Class BiMorphism (PROP1 PROP2 : bi) `{BiEmbedding PROP1 PROP2} := {
+  bi_mor_ne :> NonExpansive bi_embedding;
+  bi_mor_mono :> Proper ((⊢) ==> (⊢)) bi_embedding;
+  bi_mor_emp : ⎡emp⎤ ⊣⊢ emp;
+  bi_mor_impl_2 P Q : (⎡P⎤ → ⎡Q⎤) ⊢ ⎡P → Q⎤;
+  bi_mor_forall_2 A (Φ : A → PROP1) : (∀ x, ⎡Φ x⎤) ⊢ ⎡∀ x, Φ x⎤;
+  bi_mor_exist_1 A (Φ : A → PROP1) : ⎡∃ x, Φ x⎤ ⊢ ∃ x, ⎡Φ x⎤;
+  bi_mor_internal_eq_1 (A : ofeT) (x y : A) : ⎡x ≡ y⎤ ⊢ x ≡ y;
+  bi_mor_sep P Q : ⎡P ∗ Q⎤ ⊣⊢ ⎡P⎤ ∗ ⎡Q⎤;
+  bi_mor_wand_2 P Q : (⎡P⎤ -∗ ⎡Q⎤) ⊢ ⎡P -∗ Q⎤;
+  bi_mor_plainly P : ⎡bi_plainly P⎤ ⊣⊢ bi_plainly ⎡P⎤;
+  bi_mor_persistently P : ⎡bi_persistently P⎤ ⊣⊢ bi_persistently ⎡P⎤
 }.
 
-Class SbiMorphism {PROP1 PROP2 : sbi} (f : PROP1 → PROP2) := {
-  sbi_mor_bi_mor :> BiMorphism f;
-  sbi_mor_later P : f (▷ P) ⊣⊢ ▷ f P
+Class SbiMorphism (PROP1 PROP2 : sbi) `{BiEmbedding PROP1 PROP2} := {
+  sbi_mor_bi_mor :> BiMorphism PROP1 PROP2;
+  sbi_mor_later P : ⎡▷ P⎤ ⊣⊢ ▷ ⎡P⎤
 }.