Another failed approach to avoid declaring other projections than the carrier as canonical.

theories/algebra/cmra.v

@@ -35,92 +35,101 @@ Notation "x ≼{ n } y" := (includedN n x y)
 Instance: Params (@includedN) 4.
 Hint Extern 0 (_ ≼{_} _) => reflexivity.
 
-Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := {
+Record cmra_laws A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := {
   (* setoids *)
-  mixin_cmra_op_ne (x : A) : NonExpansive (op x);
-  mixin_cmra_pcore_ne n x y cx :
+  law_cmra_op_ne (x : A) : NonExpansive (op x);
+  law_cmra_pcore_ne n x y cx :
     x ≡{n}≡ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≡{n}≡ cy;
-  mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
+  law_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
   (* valid *)
-  mixin_cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x;
-  mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x;
+  law_cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x;
+  law_cmra_validN_S n x : ✓{S n} x → ✓{n} x;
   (* monoid *)
-  mixin_cmra_assoc : Assoc (≡) (⋅);
-  mixin_cmra_comm : Comm (≡) (⋅);
-  mixin_cmra_pcore_l x cx : pcore x = Some cx → cx ⋅ x ≡ x;
-  mixin_cmra_pcore_idemp x cx : pcore x = Some cx → pcore cx ≡ Some cx;
-  mixin_cmra_pcore_mono x y cx :
+  law_cmra_assoc : Assoc (≡) (⋅);
+  law_cmra_comm : Comm (≡) (⋅);
+  law_cmra_pcore_l x cx : pcore x = Some cx → cx ⋅ x ≡ x;
+  law_cmra_pcore_idemp x cx : pcore x = Some cx → pcore cx ≡ Some cx;
+  law_cmra_pcore_mono x y cx :
     x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy;
-  mixin_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x;
-  mixin_cmra_extend n x y1 y2 :
+  law_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x;
+  law_cmra_extend n x y1 y2 :
     ✓{n} x → x ≡{n}≡ y1 ⋅ y2 →
     ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2
 }.
 
-(** Bundeled version *)
-Structure cmraT := CMRAT' {
-  cmra_car :> Type;
-  cmra_equiv : Equiv cmra_car;
-  cmra_dist : Dist cmra_car;
-  cmra_pcore : PCore cmra_car;
-  cmra_op : Op cmra_car;
-  cmra_valid : Valid cmra_car;
-  cmra_validN : ValidN cmra_car;
-  cmra_ofe_mixin : OfeMixin cmra_car;
-  cmra_mixin : CMRAMixin cmra_car;
-  _ : Type
+Record cmra_mixin (A : Type) := CMRAMixin {
+  cmra_mixin_equiv : Equiv A;
+  cmra_mixin_dist : Dist A;
+  cmra_mixin_pcore : PCore A;
+  cmra_mixin_op : Op A;
+  cmra_mixin_valid : Valid A;
+  cmra_mixin_validN : ValidN A;
+  cmra_mixin_ofe_laws_of : ofe_laws A;
+  cmra_mixin_laws_of : cmra_laws A;
 }.
-Arguments CMRAT' _ {_ _ _ _ _ _} _ _ _.
-Notation CMRAT A m m' := (CMRAT' A m m' A).
+Arguments CMRAMixin {_ _ _ _ _ _ _} _ _.
+
+(** Bundeled version *)
+Structure cmraT := CMRAT' { cmra_car :> Type; _ : cmra_mixin cmra_car; _ : Type }.
+Notation CMRAT A m := (CMRAT' A m A).
+Add Printing Constructor cmraT.
 Arguments cmra_car : simpl never.
-Arguments cmra_equiv : simpl never.
-Arguments cmra_dist : simpl never.
+
+Definition cmra_mixin_of (A : cmraT) : cmra_mixin A := let 'CMRAT' _ m _ := A in m.
+Arguments cmra_mixin_of : simpl never.
+
+Definition cmra_pcore {A : cmraT} : PCore A := cmra_mixin_pcore _ (cmra_mixin_of A).
 Arguments cmra_pcore : simpl never.
-Arguments cmra_op : simpl never.
-Arguments cmra_valid : simpl never.
-Arguments cmra_validN : simpl never.
-Arguments cmra_ofe_mixin : simpl never.
-Arguments cmra_mixin : simpl never.
-Add Printing Constructor cmraT.
 Hint Extern 0 (PCore _) => eapply (@cmra_pcore _) : typeclass_instances.
+Definition cmra_op {A : cmraT} : Op A := cmra_mixin_op _ (cmra_mixin_of A).
+Arguments cmra_op : simpl never.
 Hint Extern 0 (Op _) => eapply (@cmra_op _) : typeclass_instances.
+Definition cmra_valid {A : cmraT} : Valid A := cmra_mixin_valid _ (cmra_mixin_of A).
+Arguments cmra_valid : simpl never.
 Hint Extern 0 (Valid _) => eapply (@cmra_valid _) : typeclass_instances.
+Definition cmra_validN {A : cmraT} : ValidN A := cmra_mixin_validN _ (cmra_mixin_of A).
+Arguments cmra_validN : simpl never.
 Hint Extern 0 (ValidN _) => eapply (@cmra_validN _) : typeclass_instances.
-Coercion cmra_ofeC (A : cmraT) : ofeT := OfeT A (cmra_ofe_mixin A).
+
+Definition cmra_ofe_mixin_of {A} (m : cmra_mixin A) : ofe_mixin A :=
+  OfeMixin (cmra_mixin_ofe_laws_of _ m).
+Coercion cmra_ofeC (A : cmraT) : ofeT :=
+  OfeT A (cmra_ofe_mixin_of (cmra_mixin_of A)).
 Canonical Structure cmra_ofeC.
 
 (** Lifting properties from the mixin *)
 Section cmra_mixin.
   Context {A : cmraT}.
   Implicit Types x y : A.
+  Local Coercion cmra_mixin_of : cmraT >-> cmra_mixin.
   Global Instance cmra_op_ne (x : A) : NonExpansive (op x).
-  Proof. apply (mixin_cmra_op_ne _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_op_ne _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_pcore_ne n x y cx :
     x ≡{n}≡ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≡{n}≡ cy.
-  Proof. apply (mixin_cmra_pcore_ne _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_pcore_ne _ (cmra_mixin_laws_of _ A)). Qed.
   Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n).
-  Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_validN_ne _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x.
-  Proof. apply (mixin_cmra_valid_validN _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_valid_validN _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_validN_S n x : ✓{S n} x → ✓{n} x.
-  Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_validN_S _ (cmra_mixin_laws_of _ A)). Qed.
   Global Instance cmra_assoc : Assoc (≡) (@op A _).
-  Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_assoc _ (cmra_mixin_laws_of _ A)). Qed.
   Global Instance cmra_comm : Comm (≡) (@op A _).
-  Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_comm _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_pcore_l x cx : pcore x = Some cx → cx ⋅ x ≡ x.
-  Proof. apply (mixin_cmra_pcore_l _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_pcore_l _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_pcore_idemp x cx : pcore x = Some cx → pcore cx ≡ Some cx.
-  Proof. apply (mixin_cmra_pcore_idemp _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_pcore_idemp _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_pcore_mono x y cx :
     x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy.
-  Proof. apply (mixin_cmra_pcore_mono _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_pcore_mono _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x.
-  Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_validN_op_l _ (cmra_mixin_laws_of _ A)). Qed.
   Lemma cmra_extend n x y1 y2 :
     ✓{n} x → x ≡{n}≡ y1 ⋅ y2 →
     ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2.
-  Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed.
+  Proof. apply (law_cmra_extend _ (cmra_mixin_laws_of _ A)). Qed.
 End cmra_mixin.
 
 Definition opM {A : cmraT} (x : A) (my : option A) :=
@@ -163,56 +172,66 @@ Arguments core' _ _ _ /.
 (** * CMRAs with a unit element *)
 (** We use the notation ∅ because for most instances (maps, sets, etc) the
 `empty' element is the unit. *)
-Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := {
+Record ucmra_laws A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := {
   mixin_ucmra_unit_valid : ✓ ∅;
   mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅);
   mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅
 }.
 
-Structure ucmraT := UCMRAT' {
-  ucmra_car :> Type;
-  ucmra_equiv : Equiv ucmra_car;
-  ucmra_dist : Dist ucmra_car;
-  ucmra_pcore : PCore ucmra_car;
-  ucmra_op : Op ucmra_car;
-  ucmra_valid : Valid ucmra_car;
-  ucmra_validN : ValidN ucmra_car;
-  ucmra_empty : Empty ucmra_car;
-  ucmra_ofe_mixin : OfeMixin ucmra_car;
-  ucmra_cmra_mixin : CMRAMixin ucmra_car;
-  ucmra_mixin : UCMRAMixin ucmra_car;
-  _ : Type;
+Record ucmra_mixin (A : Type) := UCMRAMixin {
+  ucmra_mixin_equiv : Equiv A;
+  ucmra_mixin_dist : Dist A;
+  ucmra_mixin_pcore : PCore A;
+  ucmra_mixin_op : Op A;
+  ucmra_mixin_valid : Valid A;
+  ucmra_mixin_validN : ValidN A;
+  ucmra_mixin_empty : Empty A;
+  ucmra_mixin_ofe_laws_of : ofe_laws A;
+  ucmra_mixin_cmra_laws_of : cmra_laws A;
+  ucmra_mixin_laws_of : ucmra_laws A;
 }.
-Arguments UCMRAT' _ {_ _ _ _ _ _ _} _ _ _ _.
-Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A).
-Arguments ucmra_car : simpl never.
-Arguments ucmra_equiv : simpl never.
-Arguments ucmra_dist : simpl never.
-Arguments ucmra_pcore : simpl never.
-Arguments ucmra_op : simpl never.
-Arguments ucmra_valid : simpl never.
-Arguments ucmra_validN : simpl never.
-Arguments ucmra_ofe_mixin : simpl never.
-Arguments ucmra_cmra_mixin : simpl never.
-Arguments ucmra_mixin : simpl never.
+Arguments UCMRAMixin {_ _ _ _ _ _ _ _} _ _ _.
+
+Structure ucmraT :=
+  UCMRAT' { ucmra_car :> Type; _ : ucmra_mixin ucmra_car; _ : Type }.
+Notation UCMRAT A m := (UCMRAT' A m A).
 Add Printing Constructor ucmraT.
+Arguments ucmra_car : simpl never.
+
+Definition ucmra_mixin_of (A : ucmraT) : ucmra_mixin A :=
+  let 'UCMRAT' _ m _ := A in m.
+Arguments ucmra_mixin_of : simpl never.
+
+Definition ucmra_empty {A : ucmraT} : Empty A :=
+  ucmra_mixin_empty _ (ucmra_mixin_of A).
+Arguments ucmra_empty : simpl never.
 Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances.
-Coercion ucmra_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A).
+
+Definition ucmra_ofe_mixin_of {A} (m : ucmra_mixin A) : ofe_mixin A :=
+  OfeMixin (ucmra_mixin_ofe_laws_of _ m).
+Definition ucmra_cmra_mixin_of {A} (m : ucmra_mixin A) : cmra_mixin A :=
+  CMRAMixin (ucmra_mixin_ofe_laws_of _ m) (ucmra_mixin_cmra_laws_of _ m).
+Arguments ucmra_ofe_mixin_of : simpl never.
+Arguments ucmra_cmra_mixin_of : simpl never.
+
+Coercion ucmra_ofeC (A : ucmraT) : ofeT :=
+  OfeT A (ucmra_ofe_mixin_of (ucmra_mixin_of A)).
 Canonical Structure ucmra_ofeC.
 Coercion ucmra_cmraR (A : ucmraT) : cmraT :=
-  CMRAT A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A).
+  CMRAT A (ucmra_cmra_mixin_of (ucmra_mixin_of A)).
 Canonical Structure ucmra_cmraR.
 
 (** Lifting properties from the mixin *)
 Section ucmra_mixin.
   Context {A : ucmraT}.
   Implicit Types x y : A.
+  Local Coercion ucmra_mixin_of : ucmraT >-> ucmra_mixin.
   Lemma ucmra_unit_valid : ✓ (∅ : A).
-  Proof. apply (mixin_ucmra_unit_valid _ (ucmra_mixin A)). Qed.
+  Proof. apply (mixin_ucmra_unit_valid _ (ucmra_mixin_laws_of _ A)). Qed.
   Global Instance ucmra_unit_left_id : LeftId (≡) ∅ (@op A _).
-  Proof. apply (mixin_ucmra_unit_left_id _ (ucmra_mixin A)). Qed.
+  Proof. apply (mixin_ucmra_unit_left_id _ (ucmra_mixin_laws_of _ A)). Qed.
   Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅.
-  Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed.
+  Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin_laws_of _ A)). Qed.
 End ucmra_mixin.
 
 (** * Discrete CMRAs *)
@@ -698,7 +717,7 @@ Section cmra_total.
   Context (extend : ∀ n (x y1 y2 : A),
     ✓{n} x → x ≡{n}≡ y1 ⋅ y2 →
     ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2).
-  Lemma cmra_total_mixin : CMRAMixin A.
+  Lemma cmra_total_laws : cmra_laws A.
   Proof using Type*.
     split; auto.
     - intros n x y ? Hcx%core_ne Hx; move: Hcx. rewrite /core /= Hx /=.
@@ -850,7 +869,7 @@ End cmra_transport.
 
 (** * Instances *)
 (** ** Discrete CMRA *)
-Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := {
+Record ra_laws A `{Equiv A, PCore A, Op A, Valid A} := {
   (* setoids *)
   ra_op_proper (x : A) : Proper ((≡) ==> (≡)) (op x);
   ra_core_proper x y cx :
@@ -869,18 +888,19 @@ Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := {
 Section discrete.
   Local Set Default Proof Using "Type*".
   Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A (≡)}.
-  Context (ra_mix : RAMixin A).
+  Context (laws : ra_laws A).
   Existing Instances discrete_dist.
 
   Instance discrete_validN : ValidN A := λ n x, ✓ x.
-  Definition discrete_cmra_mixin : CMRAMixin A.
+  Definition discrete_cmra_laws : cmra_laws A.
   Proof.
-    destruct ra_mix; split; try done.
+    destruct laws; split; try done.
     - intros x; split; first done. by move=> /(_ 0).
     - intros n x y1 y2 ??; by exists y1, y2.
   Qed.
 End discrete.
 
+(*
 Notation discreteR A ra_mix :=
   (CMRAT A discrete_ofe_mixin (discrete_cmra_mixin ra_mix)).
 Notation discreteUR A ra_mix ucmra_mix :=
@@ -889,7 +909,7 @@ Notation discreteUR A ra_mix ucmra_mix :=
 Global Instance discrete_cmra_discrete `{Equiv A, PCore A, Op A, Valid A,
   @Equivalence A (≡)} (ra_mix : RAMixin A) : CMRADiscrete (discreteR A ra_mix).
 Proof. split. apply _. done. Qed.
-
+*)
 Section ra_total.
   Local Set Default Proof Using "Type*".
   Context A `{Equiv A, PCore A, Op A, Valid A}.
@@ -903,7 +923,7 @@ Section ra_total.
   Context (core_idemp : ∀ x : A, core (core x) ≡ core x).
   Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y).
   Context (valid_op_l : ∀ x y : A, ✓ (x ⋅ y) → ✓ x).
-  Lemma ra_total_mixin : RAMixin A.
+  Lemma ra_total_laws : ra_laws A.
   Proof.
     split; auto.
     - intros x y ? Hcx%core_proper Hx; move: Hcx. rewrite /core /= Hx /=.
@@ -922,15 +942,17 @@ Section unit.
   Instance unit_validN : ValidN () := λ n x, True.
   Instance unit_pcore : PCore () := λ x, Some x.
   Instance unit_op : Op () := λ x y, ().
-  Lemma unit_cmra_mixin : CMRAMixin ().
-  Proof. apply discrete_cmra_mixin, ra_total_mixin; by eauto. Qed.
-  Canonical Structure unitR : cmraT := CMRAT () unit_ofe_mixin unit_cmra_mixin.
+  Lemma unit_cmra_laws : cmra_laws ().
+  Proof. apply discrete_cmra_laws, ra_total_laws; by eauto. Qed.
+  Definition unit_cmra_mixin := CMRAMixin unit_ofe_laws unit_cmra_laws.
+  Canonical Structure unitR : cmraT := CMRAT () unit_cmra_mixin.
 
   Instance unit_empty : Empty () := ().
-  Lemma unit_ucmra_mixin : UCMRAMixin ().
+  Lemma unit_ucmra_laws : ucmra_laws ().
   Proof. done. Qed.
-  Canonical Structure unitUR : ucmraT :=
-    UCMRAT () unit_ofe_mixin unit_cmra_mixin unit_ucmra_mixin.
+  Definition unit_ucmra_mixin :=
+    UCMRAMixin unit_ofe_laws unit_cmra_laws unit_ucmra_laws.
+  Canonical Structure unitUR : ucmraT := UCMRAT () unit_ucmra_mixin.
 
   Global Instance unit_cmra_discrete : CMRADiscrete unitR.
   Proof. done. Qed.
@@ -953,31 +975,35 @@ Section nat.
     - intros [z ->]; unfold op, nat_op; lia.
     - exists (y - x). by apply le_plus_minus.
   Qed.
-  Lemma nat_ra_mixin : RAMixin nat.
+  Lemma nat_ra_laws : ra_laws nat.
   Proof.
-    apply ra_total_mixin; try by eauto.
+    apply ra_total_laws; try by eauto.
     - solve_proper.
     - intros x y z. apply Nat.add_assoc.
     - intros x y. apply Nat.add_comm.
     - by exists 0.
   Qed.
+(*
+  Definition nat_cmra_mixin := CMRAMixin unit_ofe_laws unit_cmra_laws.
   Canonical Structure natR : cmraT := discreteR nat nat_ra_mixin.
-
+*)
   Instance nat_empty : Empty nat := 0.
-  Lemma nat_ucmra_mixin : UCMRAMixin nat.
+  Lemma nat_ucmra_laws : ucmra_laws nat.
   Proof. split; apply _ || done. Qed.
+(*
   Canonical Structure natUR : ucmraT :=
     discreteUR nat nat_ra_mixin nat_ucmra_mixin.
 
   Global Instance nat_cmra_discrete : CMRADiscrete natR.
   Proof. constructor; apply _ || done. Qed.
-
   Global Instance nat_cancelable (x : nat) : Cancelable x.
   Proof. by intros ???? ?%Nat.add_cancel_l. Qed.
+*)
 End nat.
 
 Definition mnat := nat.
 
+(*
 Section mnat.
   Instance mnat_valid : Valid mnat := λ x, True.
   Instance mnat_validN : ValidN mnat := λ n x, True.
@@ -1045,6 +1071,7 @@ Section positive.
     by apply leibniz_equiv.
   Qed.
 End positive.
+*)
 
 (** ** Product *)
 Section prod.
@@ -1082,7 +1109,7 @@ Section prod.
     intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto.
   Qed.
 
-  Definition prod_cmra_mixin : CMRAMixin (A * B).
+  Definition prod_cmra_laws : cmra_laws (A * B).
   Proof.
     split; try apply _.
     - by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
@@ -1111,8 +1138,8 @@ Section prod.
       destruct (cmra_extend n (x.2) (y1.2) (y2.2)) as (z21&z22&?&?&?); auto.
       by exists (z11,z21), (z12,z22).
   Qed.
-  Canonical Structure prodR :=
-    CMRAT (A * B) prod_ofe_mixin prod_cmra_mixin.
+  Definition prod_cmra_mixin := CMRAMixin prod_ofe_laws prod_cmra_laws.
+  Canonical Structure prodR := CMRAT (A * B) prod_cmra_mixin.
 
   Lemma pair_op (a a' : A) (b b' : B) : (a, b) ⋅ (a', b') = (a ⋅ a', b ⋅ b').
   Proof. done. Qed.
@@ -1152,15 +1179,16 @@ Section prod_unit.
   Context {A B : ucmraT}.
 
   Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (∅, ∅).
-  Lemma prod_ucmra_mixin : UCMRAMixin (A * B).
+  Lemma prod_ucmra_laws : ucmra_laws (A * B).
   Proof.
     split.
     - split; apply ucmra_unit_valid.
     - by split; rewrite /=left_id.
     - rewrite prod_pcore_Some'; split; apply (persistent _).
   Qed.
-  Canonical Structure prodUR :=
-    UCMRAT (A * B) prod_ofe_mixin prod_cmra_mixin prod_ucmra_mixin.
+  Definition prod_ucmra_mixin :=
+    UCMRAMixin prod_ofe_laws prod_cmra_laws prod_ucmra_laws.