move indexed products to their own file

_CoqProject

@@ -47,6 +47,7 @@ algebra/dra.v
 algebra/cofe_solver.v
 algebra/agree.v
 algebra/excl.v
+algebra/iprod.v
 algebra/functor.v
 program_logic/upred.v
 program_logic/model.v

algebra/cmra.v

@@ -535,132 +535,3 @@ Proof.
     by split; apply includedN_preserving.
   * by intros n x [??]; split; simpl; apply validN_preserving.
 Qed.
-
-(** ** Indexed product *)
-Section iprod_cmra.
-  Context {A} {B : A → cmraT}.
-  Implicit Types f g : iprod B.
-  Instance iprod_op : Op (iprod B) := λ f g x, f x ⋅ g x.
-  Definition iprod_lookup_op f g x : (f ⋅ g) x = f x ⋅ g x := eq_refl.
-  Instance iprod_unit : Unit (iprod B) := λ f x, unit (f x).
-  Definition iprod_lookup_unit f x : (unit f) x = unit (f x) := eq_refl.
-  Global Instance iprod_empty `{∀ x, Empty (B x)} : Empty (iprod B) := λ x, ∅.
-  Instance iprod_validN : ValidN (iprod B) := λ n f, ∀ x, ✓{n} (f x).
-  Instance iprod_minus : Minus (iprod B) := λ f g x, f x ⩪ g x.
-  Definition iprod_lookup_minus f g x : (f ⩪ g) x = f x ⩪ g x := eq_refl.
-  Lemma iprod_includedN_spec (f g : iprod B) n : f ≼{n} g ↔ ∀ x, f x ≼{n} g x.
-  Proof.
-    split.
-    * by intros [h Hh] x; exists (h x); rewrite /op /iprod_op (Hh x).
-    * intros Hh; exists (g ⩪ f)=> x; specialize (Hh x).
-      by rewrite /op /iprod_op /minus /iprod_minus cmra_op_minus.
-  Qed.
-  Definition iprod_cmra_mixin : CMRAMixin (iprod B).
-  Proof.
-    split.
-    * by intros n f1 f2 f3 Hf x; rewrite iprod_lookup_op (Hf x).
-    * by intros n f1 f2 Hf x; rewrite iprod_lookup_unit (Hf x).
-    * by intros n f1 f2 Hf ? x; rewrite -(Hf x).
-    * by intros n f f' Hf g g' Hg i; rewrite iprod_lookup_minus (Hf i) (Hg i).
-    * by intros f x.
-    * intros n f Hf x; apply cmra_validN_S, Hf.
-    * by intros f1 f2 f3 x; rewrite iprod_lookup_op associative.
-    * by intros f1 f2 x; rewrite iprod_lookup_op commutative.
-    * by intros f x; rewrite iprod_lookup_op iprod_lookup_unit cmra_unit_l.
-    * by intros f x; rewrite iprod_lookup_unit cmra_unit_idempotent.
-    * intros n f1 f2; rewrite !iprod_includedN_spec=> Hf x.
-      by rewrite iprod_lookup_unit; apply cmra_unit_preservingN, Hf.
-    * intros n f1 f2 Hf x; apply cmra_validN_op_l with (f2 x), Hf.
-    * intros n f1 f2; rewrite iprod_includedN_spec=> Hf x.
-      by rewrite iprod_lookup_op iprod_lookup_minus cmra_op_minus; try apply Hf.
-  Qed.
-  Definition iprod_cmra_extend_mixin : CMRAExtendMixin (iprod B).
-  Proof.
-    intros n f f1 f2 Hf Hf12.
-    set (g x := cmra_extend_op n (f x) (f1 x) (f2 x) (Hf x) (Hf12 x)).
-    exists ((λ x, (proj1_sig (g x)).1), (λ x, (proj1_sig (g x)).2)).
-    split_ands; intros x; apply (proj2_sig (g x)).
-  Qed.
-  Canonical Structure iprodRA : cmraT :=
-    CMRAT iprod_cofe_mixin iprod_cmra_mixin iprod_cmra_extend_mixin.
-  Global Instance iprod_cmra_identity `{∀ x, Empty (B x)} :
-    (∀ x, CMRAIdentity (B x)) → CMRAIdentity iprodRA.
-  Proof.
-    intros ?; split.
-    * intros n x; apply cmra_empty_valid.
-    * by intros f x; rewrite iprod_lookup_op left_id.
-    * by intros f Hf x; apply (timeless _).
-  Qed.
-
-  Context `{∀ x x' : A, Decision (x = x')}.
-  Lemma iprod_insert_updateP x (P : B x → Prop) (Q : iprod B → Prop) g y1 :
-    y1 ~~>: P → (∀ y2, P y2 → Q (iprod_insert x y2 g)) →
-    iprod_insert x y1 g ~~>: Q.
-  Proof.
-    intros Hy1 HP gf n Hg. destruct (Hy1 (gf x) n) as (y2&?&?).
-    { move: (Hg x). by rewrite iprod_lookup_op iprod_lookup_insert. }
-    exists (iprod_insert x y2 g); split; [auto|].
-    intros x'; destruct (decide (x' = x)) as [->|];
-      rewrite iprod_lookup_op ?iprod_lookup_insert //; [].
-    move: (Hg x'). by rewrite iprod_lookup_op !iprod_lookup_insert_ne.
-  Qed.
-
-  Lemma iprod_insert_updateP' x (P : B x → Prop) g y1 :
-    y1 ~~>: P →
-    iprod_insert x y1 g ~~>: λ g', ∃ y2, g' = iprod_insert x y2 g ∧ P y2.
-  Proof. eauto using iprod_insert_updateP. Qed.
-  Lemma iprod_insert_update g x y1 y2 :
-
-    y1 ~~> y2 → iprod_insert x y1 g ~~> iprod_insert x y2 g.
-  Proof.
-    rewrite !cmra_update_updateP;
-      eauto using iprod_insert_updateP with congruence.
-  Qed.
-
-  Context `{∀ x, Empty (B x)} `{∀ x, CMRAIdentity (B x)}.
-  Lemma iprod_op_singleton (x : A) (y1 y2 : B x) :
-    iprod_singleton x y1 ⋅ iprod_singleton x y2 ≡ iprod_singleton x (y1 ⋅ y2).
-  Proof.
-    intros x'; destruct (decide (x' = x)) as [->|].
-    * by rewrite iprod_lookup_op !iprod_lookup_singleton.
-    * by rewrite iprod_lookup_op !iprod_lookup_singleton_ne // left_id.
-  Qed.
-
-  Lemma iprod_singleton_updateP x (P : B x → Prop) (Q : iprod B → Prop) y1 :
-    y1 ~~>: P → (∀ y2, P y2 → Q (iprod_singleton x y2)) →
-    iprod_singleton x y1 ~~>: Q.
-  Proof. rewrite /iprod_singleton; eauto using iprod_insert_updateP. Qed.
-
-  Lemma iprod_singleton_updateP' x (P : B x → Prop) y1 :
-    y1 ~~>: P →
-    iprod_singleton x y1 ~~>: λ g', ∃ y2, g' = iprod_singleton x y2 ∧ P y2.
-  Proof. eauto using iprod_singleton_updateP. Qed.
-
-  Lemma iprod_singleton_updateP_empty x (P : B x → Prop) (Q : iprod B → Prop) :
-    (∅ ~~>: P) → (∀ y2, P y2 → Q (iprod_singleton x y2)) →
-    ∅ ~~>: Q.
-  Proof.
-    intros Hx HQ gf n Hg. destruct (Hx (gf x) n) as (y2&?&?).
-    { apply: Hg. }
-    exists (iprod_singleton x y2).
-    split; first by apply HQ.
-    intros x'; destruct (decide (x' = x)) as [->|];
-      rewrite iprod_lookup_op /iprod_singleton ?iprod_lookup_insert //; [].
-    move:(Hg x'). by rewrite iprod_lookup_insert_ne // left_id.
-  Qed.
-
-  Lemma iprod_singleton_update x y1 y2 :
-    y1 ~~> y2 → iprod_singleton x y1 ~~> iprod_singleton x y2.
-  Proof. by intros; apply iprod_insert_update. Qed.
-End iprod_cmra.
-
-Arguments iprodRA {_} _.
-
-Instance iprod_map_cmra_monotone {A} {B1 B2: A → cmraT} (f : ∀ x, B1 x → B2 x) :
-  (∀ x, CMRAMonotone (f x)) → CMRAMonotone (iprod_map f).
-Proof.
-  split.
-  * intros n g1 g2; rewrite !iprod_includedN_spec=> Hf x.
-    rewrite /iprod_map; apply includedN_preserving, Hf.
-  * intros n g Hg x; rewrite /iprod_map; apply validN_preserving, Hg.
-Qed.

algebra/cofe.v

@@ -367,95 +367,3 @@ Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B :=
   CofeMor (later_map f).
 Instance laterC_map_contractive (A B : cofeT) : Contractive (@laterC_map A B).
 Proof. intros n f g Hf n'; apply Hf. Qed.
-
-(** Indexed product *)
-(** Need to put this in a definition to make canonical structures to work. *)
-Definition iprod {A} (B : A → cofeT) := ∀ x, B x.
-Definition iprod_insert `{∀ x x' : A, Decision (x = x')} {B : A → cofeT}
-    (x : A) (y : B x) (f : iprod B) : iprod B := λ x',
-  match decide (x = x') with left H => eq_rect _ B y _ H | right _ => f x' end.
-Definition iprod_singleton
-    `{∀ x x' : A, Decision (x = x')} {B : A → cofeT} `{∀ x : A, Empty (B x)}
-  (x : A) (y : B x) : iprod B := iprod_insert x y (λ _, ∅).
-
-Section iprod_cofe.
-  Context {A} {B : A → cofeT}.
-  Implicit Types x : A.
-  Implicit Types f g : iprod B.
-  Instance iprod_equiv : Equiv (iprod B) := λ f g, ∀ x, f x ≡ g x.
-  Instance iprod_dist : Dist (iprod B) := λ n f g, ∀ x, f x ={n}= g x.
-  Program Definition iprod_chain (c : chain (iprod B)) (x : A) : chain (B x) :=
-    {| chain_car n := c n x |}.
-  Next Obligation. by intros c x n i ?; apply (chain_cauchy c). Qed.
-  Program Instance iprod_compl : Compl (iprod B) := λ c x,
-    compl (iprod_chain c x).
-  Definition iprod_cofe_mixin : CofeMixin (iprod B).
-  Proof.
-    split.
-    * intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
-      intros Hfg k; apply equiv_dist; intros n; apply Hfg.
-    * intros n; split.
-      + by intros f x.
-      + by intros f g ? x.
-      + by intros f g h ?? x; transitivity (g x).
-    * intros n f g Hfg x; apply dist_S, Hfg.
-    * by intros f g x.
-    * intros c n x.
-      rewrite /compl /iprod_compl (conv_compl (iprod_chain c x) n).
-      apply (chain_cauchy c); lia.
-  Qed.
-  Canonical Structure iprodC : cofeT := CofeT iprod_cofe_mixin.
-
-  Context `{∀ x x' : A, Decision (x = x')}.
-  Global Instance iprod_insert_ne x n :
-    Proper (dist n ==> dist n ==> dist n) (iprod_insert x).
-  Proof.
-    intros y1 y2 ? f1 f2 ? x'; rewrite /iprod_insert.
-    by destruct (decide _) as [[]|].
-  Qed.
-  Global Instance iprod_insert_proper x :
-    Proper ((≡) ==> (≡) ==> (≡)) (iprod_insert x) := ne_proper_2 _.
-  Lemma iprod_lookup_insert f x y : (iprod_insert x y f) x = y.
-  Proof.
-    rewrite /iprod_insert; destruct (decide _) as [Hx|]; last done.
-    by rewrite (proof_irrel Hx eq_refl).
-  Qed.
-  Lemma iprod_lookup_insert_ne f x x' y :
-    x ≠ x' → (iprod_insert x y f) x' = f x'.
-  Proof. by rewrite /iprod_insert; destruct (decide _). Qed.
-
-  Context `{∀ x : A, Empty (B x)}.
-  Global Instance iprod_singleton_ne x n :
-    Proper (dist n ==> dist n) (iprod_singleton x).
-  Proof. by intros y1 y2 Hy; rewrite /iprod_singleton Hy. Qed.
-  Global Instance iprod_singleton_proper x :
-    Proper ((≡) ==> (≡)) (iprod_singleton x) := ne_proper _.
-  Lemma iprod_lookup_singleton x y : (iprod_singleton x y) x = y.
-  Proof. by rewrite /iprod_singleton iprod_lookup_insert. Qed.
-  Lemma iprod_lookup_singleton_ne x x' y :
-    x ≠ x' → (iprod_singleton x y) x' = ∅.
-  Proof. intros; by rewrite /iprod_singleton iprod_lookup_insert_ne. Qed.
-End iprod_cofe.
-
-Arguments iprodC {_} _.
-
-Definition iprod_map {A} {B1 B2 : A → cofeT} (f : ∀ x, B1 x → B2 x)
-  (g : iprod B1) : iprod B2 := λ x, f _ (g x).
-Lemma iprod_map_ext {A} {B1 B2 : A → cofeT} (f1 f2 : ∀ x, B1 x → B2 x) g :
-  (∀ x, f1 x (g x) ≡ f2 x (g x)) → iprod_map f1 g ≡ iprod_map f2 g.
-Proof. done. Qed.
-Lemma iprod_map_id {A} {B: A → cofeT} (g : iprod B) : iprod_map (λ _, id) g = g.
-Proof. done. Qed.
-Lemma iprod_map_compose {A} {B1 B2 B3 : A → cofeT}
-    (f1 : ∀ x, B1 x → B2 x) (f2 : ∀ x, B2 x → B3 x) (g : iprod B1) :
-  iprod_map (λ x, f2 x ∘ f1 x) g = iprod_map f2 (iprod_map f1 g).
-Proof. done. Qed.
-Instance iprod_map_ne {A} {B1 B2 : A → cofeT} (f : ∀ x, B1 x → B2 x) n :
-  (∀ x, Proper (dist n ==> dist n) (f x)) →
-  Proper (dist n ==> dist n) (iprod_map f).
-Proof. by intros ? y1 y2 Hy x; rewrite /iprod_map (Hy x). Qed.
-Definition iprodC_map {A} {B1 B2 : A → cofeT} (f : iprod (λ x, B1 x -n> B2 x)) :
-  iprodC B1 -n> iprodC B2 := CofeMor (iprod_map f).
-Instance iprodC_map_ne {A} {B1 B2 : A → cofeT} n :
-  Proper (dist n ==> dist n) (@iprodC_map A B1 B2).
-Proof. intros f1 f2 Hf g x; apply Hf. Qed.

algebra/functor.v

@@ -39,19 +39,3 @@ Next Obligation. by intros Σ1 Σ2 A [??]; rewrite /= !ifunctor_map_id. Qed.
 Next Obligation.
   by intros Σ1 Σ2 A B C f g [??]; rewrite /= !ifunctor_map_compose.
 Qed.
-
-Program Definition iprodF {A} (Σ : A → iFunctor) : iFunctor := {|
-  ifunctor_car B := iprodRA (λ x, Σ x B);
-  ifunctor_map B1 B2 f := iprodC_map (λ x, ifunctor_map (Σ x) f);
-|}.
-Next Obligation.
-  by intros A Σ B1 B2 n f f' ? g; apply iprodC_map_ne=>x; apply ifunctor_map_ne.
-Qed.
-Next Obligation.
-  intros A Σ B g. rewrite /= -{2}(iprod_map_id g).
-  apply iprod_map_ext=> x; apply ifunctor_map_id.
-Qed.
-Next Obligation.
-  intros A Σ B1 B2 B3 f1 f2 g. rewrite /= -iprod_map_compose.
-  apply iprod_map_ext=> y; apply ifunctor_map_compose.
-Qed.

algebra/iprod.v

+Require Export algebra.cmra.
+Require Import algebra.functor.
+
+(** Indexed product *)
+(** Need to put this in a definition to make canonical structures to work. *)
+Definition iprod {A} (B : A → cofeT) := ∀ x, B x.
+Definition iprod_insert `{∀ x x' : A, Decision (x = x')} {B : A → cofeT}
+    (x : A) (y : B x) (f : iprod B) : iprod B := λ x',
+  match decide (x = x') with left H => eq_rect _ B y _ H | right _ => f x' end.
+Definition iprod_singleton
+    `{∀ x x' : A, Decision (x = x')} {B : A → cofeT} `{∀ x : A, Empty (B x)}
+  (x : A) (y : B x) : iprod B := iprod_insert x y (λ _, ∅).
+
+Section iprod_cofe.
+  Context {A} {B : A → cofeT}.
+  Implicit Types x : A.
+  Implicit Types f g : iprod B.
+  Instance iprod_equiv : Equiv (iprod B) := λ f g, ∀ x, f x ≡ g x.
+  Instance iprod_dist : Dist (iprod B) := λ n f g, ∀ x, f x ={n}= g x.
+  Program Definition iprod_chain (c : chain (iprod B)) (x : A) : chain (B x) :=
+    {| chain_car n := c n x |}.
+  Next Obligation. by intros c x n i ?; apply (chain_cauchy c). Qed.
+  Program Instance iprod_compl : Compl (iprod B) := λ c x,
+    compl (iprod_chain c x).
+  Definition iprod_cofe_mixin : CofeMixin (iprod B).
+  Proof.
+    split.
+    * intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
+      intros Hfg k; apply equiv_dist; intros n; apply Hfg.
+    * intros n; split.
+      + by intros f x.
+      + by intros f g ? x.
+      + by intros f g h ?? x; transitivity (g x).
+    * intros n f g Hfg x; apply dist_S, Hfg.
+    * by intros f g x.
+    * intros c n x.
+      rewrite /compl /iprod_compl (conv_compl (iprod_chain c x) n).
+      apply (chain_cauchy c); lia.
+  Qed.
+  Canonical Structure iprodC : cofeT := CofeT iprod_cofe_mixin.
+
+  Context `{∀ x x' : A, Decision (x = x')}.
+  Global Instance iprod_insert_ne x n :
+    Proper (dist n ==> dist n ==> dist n) (iprod_insert x).
+  Proof.
+    intros y1 y2 ? f1 f2 ? x'; rewrite /iprod_insert.
+    by destruct (decide _) as [[]|].
+  Qed.
+  Global Instance iprod_insert_proper x :
+    Proper ((≡) ==> (≡) ==> (≡)) (iprod_insert x) := ne_proper_2 _.
+  Lemma iprod_lookup_insert f x y : (iprod_insert x y f) x = y.
+  Proof.
+    rewrite /iprod_insert; destruct (decide _) as [Hx|]; last done.
+    by rewrite (proof_irrel Hx eq_refl).
+  Qed.
+  Lemma iprod_lookup_insert_ne f x x' y :
+    x ≠ x' → (iprod_insert x y f) x' = f x'.
+  Proof. by rewrite /iprod_insert; destruct (decide _). Qed.
+
+  Context `{∀ x : A, Empty (B x)}.
+  Global Instance iprod_singleton_ne x n :
+    Proper (dist n ==> dist n) (iprod_singleton x).
+  Proof. by intros y1 y2 Hy; rewrite /iprod_singleton Hy. Qed.
+  Global Instance iprod_singleton_proper x :
+    Proper ((≡) ==> (≡)) (iprod_singleton x) := ne_proper _.
+  Lemma iprod_lookup_singleton x y : (iprod_singleton x y) x = y.
+  Proof. by rewrite /iprod_singleton iprod_lookup_insert. Qed.
+  Lemma iprod_lookup_singleton_ne x x' y :
+    x ≠ x' → (iprod_singleton x y) x' = ∅.
+  Proof. intros; by rewrite /iprod_singleton iprod_lookup_insert_ne. Qed.
+End iprod_cofe.
+
+Arguments iprodC {_} _.
+
+Definition iprod_map {A} {B1 B2 : A → cofeT} (f : ∀ x, B1 x → B2 x)
+  (g : iprod B1) : iprod B2 := λ x, f _ (g x).
+Lemma iprod_map_ext {A} {B1 B2 : A → cofeT} (f1 f2 : ∀ x, B1 x → B2 x) g :
+  (∀ x, f1 x (g x) ≡ f2 x (g x)) → iprod_map f1 g ≡ iprod_map f2 g.
+Proof. done. Qed.
+Lemma iprod_map_id {A} {B: A → cofeT} (g : iprod B) : iprod_map (λ _, id) g = g.
+Proof. done. Qed.
+Lemma iprod_map_compose {A} {B1 B2 B3 : A → cofeT}
+    (f1 : ∀ x, B1 x → B2 x) (f2 : ∀ x, B2 x → B3 x) (g : iprod B1) :
+  iprod_map (λ x, f2 x ∘ f1 x) g = iprod_map f2 (iprod_map f1 g).
+Proof. done. Qed.
+Instance iprod_map_ne {A} {B1 B2 : A → cofeT} (f : ∀ x, B1 x → B2 x) n :
+  (∀ x, Proper (dist n ==> dist n) (f x)) →
+  Proper (dist n ==> dist n) (iprod_map f).
+Proof. by intros ? y1 y2 Hy x; rewrite /iprod_map (Hy x). Qed.
+Definition iprodC_map {A} {B1 B2 : A → cofeT} (f : iprod (λ x, B1 x -n> B2 x)) :
+  iprodC B1 -n> iprodC B2 := CofeMor (iprod_map f).
+Instance iprodC_map_ne {A} {B1 B2 : A → cofeT} n :
+  Proper (dist n ==> dist n) (@iprodC_map A B1 B2).
+Proof. intros f1 f2 Hf g x; apply Hf. Qed.
+
+Section iprod_cmra.
+  Context {A} {B : A → cmraT}.
+  Implicit Types f g : iprod B.
+  Instance iprod_op : Op (iprod B) := λ f g x, f x ⋅ g x.
+  Definition iprod_lookup_op f g x : (f ⋅ g) x = f x ⋅ g x := eq_refl.
+  Instance iprod_unit : Unit (iprod B) := λ f x, unit (f x).
+  Definition iprod_lookup_unit f x : (unit f) x = unit (f x) := eq_refl.
+  Global Instance iprod_empty `{∀ x, Empty (B x)} : Empty (iprod B) := λ x, ∅.
+  Instance iprod_validN : ValidN (iprod B) := λ n f, ∀ x, ✓{n} (f x).
+  Instance iprod_minus : Minus (iprod B) := λ f g x, f x ⩪ g x.
+  Definition iprod_lookup_minus f g x : (f ⩪ g) x = f x ⩪ g x := eq_refl.
+  Lemma iprod_includedN_spec (f g : iprod B) n : f ≼{n} g ↔ ∀ x, f x ≼{n} g x.
+  Proof.
+    split.
+    * by intros [h Hh] x; exists (h x); rewrite /op /iprod_op (Hh x).
+    * intros Hh; exists (g ⩪ f)=> x; specialize (Hh x).
+      by rewrite /op /iprod_op /minus /iprod_minus cmra_op_minus.
+  Qed.
+  Definition iprod_cmra_mixin : CMRAMixin (iprod B).
+  Proof.
+    split.
+    * by intros n f1 f2 f3 Hf x; rewrite iprod_lookup_op (Hf x).
+    * by intros n f1 f2 Hf x; rewrite iprod_lookup_unit (Hf x).
+    * by intros n f1 f2 Hf ? x; rewrite -(Hf x).
+    * by intros n f f' Hf g g' Hg i; rewrite iprod_lookup_minus (Hf i) (Hg i).
+    * by intros f x.
+    * intros n f Hf x; apply cmra_validN_S, Hf.
+    * by intros f1 f2 f3 x; rewrite iprod_lookup_op associative.
+    * by intros f1 f2 x; rewrite iprod_lookup_op commutative.
+    * by intros f x; rewrite iprod_lookup_op iprod_lookup_unit cmra_unit_l.
+    * by intros f x; rewrite iprod_lookup_unit cmra_unit_idempotent.
+    * intros n f1 f2; rewrite !iprod_includedN_spec=> Hf x.
+      by rewrite iprod_lookup_unit; apply cmra_unit_preservingN, Hf.
+    * intros n f1 f2 Hf x; apply cmra_validN_op_l with (f2 x), Hf.
+    * intros n f1 f2; rewrite iprod_includedN_spec=> Hf x.
+      by rewrite iprod_lookup_op iprod_lookup_minus cmra_op_minus; try apply Hf.
+  Qed.
+  Definition iprod_cmra_extend_mixin : CMRAExtendMixin (iprod B).
+  Proof.
+    intros n f f1 f2 Hf Hf12.
+    set (g x := cmra_extend_op n (f x) (f1 x) (f2 x) (Hf x) (Hf12 x)).
+    exists ((λ x, (proj1_sig (g x)).1), (λ x, (proj1_sig (g x)).2)).
+    split_ands; intros x; apply (proj2_sig (g x)).
+  Qed.
+  Canonical Structure iprodRA : cmraT :=
+    CMRAT iprod_cofe_mixin iprod_cmra_mixin iprod_cmra_extend_mixin.
+  Global Instance iprod_cmra_identity `{∀ x, Empty (B x)} :
+    (∀ x, CMRAIdentity (B x)) → CMRAIdentity iprodRA.
+  Proof.
+    intros ?; split.
+    * intros n x; apply cmra_empty_valid.
+    * by intros f x; rewrite iprod_lookup_op left_id.
+    * by intros f Hf x; apply (timeless _).
+  Qed.
+
+  Context `{∀ x x' : A, Decision (x = x')}.
+  Lemma iprod_insert_updateP x (P : B x → Prop) (Q : iprod B → Prop) g y1 :
+    y1 ~~>: P → (∀ y2, P y2 → Q (iprod_insert x y2 g)) →
+    iprod_insert x y1 g ~~>: Q.
+  Proof.
+    intros Hy1 HP gf n Hg. destruct (Hy1 (gf x) n) as (y2&?&?).
+    { move: (Hg x). by rewrite iprod_lookup_op iprod_lookup_insert. }
+    exists (iprod_insert x y2 g); split; [auto|].
+    intros x'; destruct (decide (x' = x)) as [->|];
+      rewrite iprod_lookup_op ?iprod_lookup_insert //; [].
+    move: (Hg x'). by rewrite iprod_lookup_op !iprod_lookup_insert_ne.
+  Qed.
+
+  Lemma iprod_insert_updateP' x (P : B x → Prop) g y1 :
+    y1 ~~>: P →
+    iprod_insert x y1 g ~~>: λ g', ∃ y2, g' = iprod_insert x y2 g ∧ P y2.
+  Proof. eauto using iprod_insert_updateP. Qed.
+  Lemma iprod_insert_update g x y1 y2 :
+
+    y1 ~~> y2 → iprod_insert x y1 g ~~> iprod_insert x y2 g.
+  Proof.
+    rewrite !cmra_update_updateP;
+      eauto using iprod_insert_updateP with congruence.
+  Qed.
+
+  Context `{∀ x, Empty (B x)} `{∀ x, CMRAIdentity (B x)}.
+  Lemma iprod_op_singleton (x : A) (y1 y2 : B x) :
+    iprod_singleton x y1 ⋅ iprod_singleton x y2 ≡ iprod_singleton x (y1 ⋅ y2).
+  Proof.
+    intros x'; destruct (decide (x' = x)) as [->|].
+    * by rewrite iprod_lookup_op !iprod_lookup_singleton.
+    * by rewrite iprod_lookup_op !iprod_lookup_singleton_ne // left_id.
+  Qed.
+
+  Lemma iprod_singleton_updateP x (P : B x → Prop) (Q : iprod B → Prop) y1 :
+    y1 ~~>: P → (∀ y2, P y2 → Q (iprod_singleton x y2)) →
+    iprod_singleton x y1 ~~>: Q.
+  Proof. rewrite /iprod_singleton; eauto using iprod_insert_updateP. Qed.
+
+  Lemma iprod_singleton_updateP' x (P : B x → Prop) y1 :
+    y1 ~~>: P →
+    iprod_singleton x y1 ~~>: λ g', ∃ y2, g' = iprod_singleton x y2 ∧ P y2.
+  Proof. eauto using iprod_singleton_updateP. Qed.
+
+  Lemma iprod_singleton_updateP_empty x (P : B x → Prop) (Q : iprod B → Prop) :
+    (∅ ~~>: P) → (∀ y2, P y2 → Q (iprod_singleton x y2)) →
+    ∅ ~~>: Q.
+  Proof.
+    intros Hx HQ gf n Hg. destruct (Hx (gf x) n) as (y2&?&?).
+    { apply: Hg. }
+    exists (iprod_singleton x y2).
+    split; first by apply HQ.
+    intros x'; destruct (decide (x' = x)) as [->|];
+      rewrite iprod_lookup_op /iprod_singleton ?iprod_lookup_insert //; [].
+    move:(Hg x'). by rewrite iprod_lookup_insert_ne // left_id.
+  Qed.
+
+  Lemma iprod_singleton_update x y1 y2 :
+    y1 ~~> y2 → iprod_singleton x y1 ~~> iprod_singleton x y2.
+  Proof. by intros; apply iprod_insert_update. Qed.
+End iprod_cmra.
+
+Arguments iprodRA {_} _.
+
+Instance iprod_map_cmra_monotone {A} {B1 B2: A → cmraT} (f : ∀ x, B1 x → B2 x) :
+  (∀ x, CMRAMonotone (f x)) → CMRAMonotone (iprod_map f).
+Proof.
+  split.
+  * intros n g1 g2; rewrite !iprod_includedN_spec=> Hf x.
+    rewrite /iprod_map; apply includedN_preserving, Hf.
+  * intros n g Hg x; rewrite /iprod_map; apply validN_preserving, Hg.
+Qed.
+
+Program Definition iprodF {A} (Σ : A → iFunctor) : iFunctor := {|
+  ifunctor_car B := iprodRA (λ x, Σ x B);
+  ifunctor_map B1 B2 f := iprodC_map (λ x, ifunctor_map (Σ x) f);
+|}.
+Next Obligation.
+  by intros A Σ B1 B2 n f f' ? g; apply iprodC_map_ne=>x; apply ifunctor_map_ne.
+Qed.
+Next Obligation.
+  intros A Σ B g. rewrite /= -{2}(iprod_map_id g).
+  apply iprod_map_ext=> x; apply ifunctor_map_id.
+Qed.
+Next Obligation.
+  intros A Σ B1 B2 B3 f1 f2 g. rewrite /= -iprod_map_compose.
+  apply iprod_map_ext=> y; apply ifunctor_map_compose.
+Qed.