From fri.algebra Require Export cmra. From stdpp Require Export gmap. From fri.algebra Require Import upred upred_bi updates local_updates. Section cofe. Context `{Countable K} {A : ofeT}. Implicit Types m : gmap K A. Instance gmap_dist : Dist (gmap K A) := λ n m1 m2, ∀ i, m1 !! i ≡{n}≡ m2 !! i. Definition gmap_cofe_mixin : OfeMixin (gmap K A). Proof. split. - intros m1 m2; split. + by intros Hm n k; apply equiv_dist. + intros Hm k; apply equiv_dist; intros n; apply Hm. - intros n; split. + by intros m k. + by intros m1 m2 ? k. + by intros m1 m2 m3 ?? k; trans (m2 !! k). - by intros n m1 m2 ? k; apply dist_S. Qed. Canonical Structure gmapO : ofeT := OfeT (gmap K A) gmap_cofe_mixin. Global Instance gmap_discrete : OfeDiscrete A → OfeDiscrete gmapO. Proof. intros ? m m' ? i. by apply (discrete _). Qed. (* why doesn't this go automatic? *) Global Instance gmapO_leibniz: LeibnizEquiv A → LeibnizEquiv gmapO. Proof. intros; change (LeibnizEquiv (gmap K A)); apply _. Qed. Global Instance lookup_ne n k : Proper (dist n ==> dist n) (lookup k : gmap K A → option A). Proof. by intros m1 m2. Qed. Global Instance lookup_proper k : Proper ((≡) ==> (≡)) (lookup k : gmap K A → option A) := _. Global Instance alter_ne f k n : Proper (dist n ==> dist n) f → Proper (dist n ==> dist n) (alter f k). Proof. intros ? m m' Hm k'. by destruct (decide (k = k')); simplify_map_eq; rewrite (Hm k'). Qed. Global Instance insert_ne i n : Proper (dist n ==> dist n ==> dist n) (insert (M:=gmap K A) i). Proof. intros x y ? m m' ? j; destruct (decide (i = j)); simplify_map_eq; [by constructor|by apply lookup_ne]. Qed. Global Instance singleton_ne i n : Proper (dist n ==> dist n) (singletonM i : A → gmap K A). Proof. by intros ???; apply insert_ne. Qed. Global Instance delete_ne i n : Proper (dist n ==> dist n) (delete (M:=gmap K A) i). Proof. intros m m' ? j; destruct (decide (i = j)); simplify_map_eq; [by constructor|by apply lookup_ne]. Qed. Global Instance gmap_empty_timeless : Discrete (∅ : gmap K A). Proof. intros m Hm i; specialize (Hm i); rewrite lookup_empty in Hm |- *. inversion_clear Hm; constructor. Qed. Global Instance gmap_lookup_timeless m i : Discrete m → Discrete (m !! i). Proof. intros ? [x|] Hx; [|by symmetry; apply: discrete]. assert (m ≡{0}≡ <[i:=x]> m) by (by symmetry in Hx; inversion Hx; ofe_subst; rewrite insert_id). by rewrite (discrete m (<[i:=x]>m)) // lookup_insert. Qed. Global Instance gmap_insert_timeless m i x : Discrete x → Discrete m → Discrete (<[i:=x]>m). Proof. intros ?? m' Hm j; destruct (decide (i = j)); simplify_map_eq. { by apply: discrete; rewrite -Hm lookup_insert. } by apply: discrete; rewrite -Hm lookup_insert_ne. Qed. Global Instance gmap_singleton_timeless i x : Discrete x → Discrete ({[ i := x ]} : gmap K A) := _. End cofe. Arguments gmapO _ {_ _} _. (* CMRA *) Section cmra. Context `{Countable K} {A : cmraT}. Implicit Types m : gmap K A. Instance gmap_op : Op (gmap K A) := merge op. Instance gmap_pcore : PCore (gmap K A) := λ m, Some (omap pcore m). Instance gmap_valid : Valid (gmap K A) := λ m, ∀ i, ✓ (m !! i). Instance gmap_validN : ValidN (gmap K A) := λ n m, ∀ i, ✓{n} (m !! i). Instance gmap_stepN : StepN (gmap K A) := λ n m1 m2, ∀ i, (m1 !! i) ⤳_(n) (m2 !! i). Lemma lookup_op m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i. Proof. by apply lookup_merge. Qed. Lemma lookup_core m i : core m !! i = core (m !! i). Proof. by apply lookup_omap. Qed. Lemma lookup_included (m1 m2 : gmap K A) : m1 ≼ m2 ↔ ∀ i, m1 !! i ≼ m2 !! i. Proof. split; [by intros [m Hm] i; exists (m !! i); rewrite -lookup_op Hm|]. revert m2. induction m1 as [|i x m Hi IH] using map_ind=> m2 Hm. { exists m2. by rewrite left_id. } destruct (IH (delete i m2)) as [m2' Hm2']. { intros j. move: (Hm j); destruct (decide (i = j)) as [->|]. - intros _. rewrite Hi. apply: ucmra_unit_least. - rewrite lookup_insert_ne // lookup_delete_ne //. } destruct (Hm i) as [my Hi']; simplify_map_eq. exists (partial_alter (λ _, my) i m2')=>j; destruct (decide (i = j)) as [->|]. - by rewrite Hi' lookup_op lookup_insert lookup_partial_alter. - move: (Hm2' j). by rewrite !lookup_op lookup_delete_ne // lookup_insert_ne // lookup_partial_alter_ne. Qed. Lemma gmap_cmra_mixin : CMRAMixin (gmap K A). Proof. apply cmra_total_mixin. - eauto. - intros n m1 m2 m3 Hm i; by rewrite !lookup_op (Hm i). - intros n m1 m2 Hm i; by rewrite !lookup_core (Hm i). - intros n m1 m2 Hm ? i; by rewrite -(Hm i). - intros m; split. + by intros ? n i; apply cmra_valid_validN. + intros Hm i; apply cmra_valid_validN=> n; apply Hm. - intros n m Hm i; apply cmra_validN_S, Hm. - intros n m1 m1' Hm1 m2 m2' Hm2 Hs i. specialize (Hs i); rewrite -Hm1 -Hm2 in Hs *; eauto. - intros n m1 m1' Hs i. specialize (Hs i); eapply cmra_stepN_S; eauto. - by intros m1 m2 m3 i; rewrite !lookup_op assoc. - by intros m1 m2 i; rewrite !lookup_op comm. - intros m i. by rewrite lookup_op lookup_core cmra_core_l. - intros m i. by rewrite !lookup_core cmra_core_idemp. - intros m1 m2; rewrite !lookup_included=> Hm i. rewrite !lookup_core. by apply cmra_core_mono. - intros n x y Hval i. rewrite !lookup_core ?lookup_op !lookup_core cmra_core_distrib //=. specialize (Hval i). rewrite !lookup_op in Hval; auto. - intros n m1 m2 Hm i; apply cmra_validN_op_l with (m2 !! i). by rewrite -lookup_op. - intros n m m1 m2 Hm Hm12. assert (∀ i, m !! i ≡{n}≡ m1 !! i ⋅ m2 !! i) as Hm12' by (by intros i; rewrite -lookup_op). set (f i := cmra_extend n (m !! i) (m1 !! i) (m2 !! i) (Hm i) (Hm12' i)). set (f_proj i := proj1_sig (f i)). exists (map_imap (λ i _, (f_proj i).1) m, map_imap (λ i _, (f_proj i).2) m); repeat split; intros i; rewrite /= ?lookup_op !map_lookup_imap. + destruct (m !! i) as [x|] eqn:Hx; rewrite !Hx /=; [|constructor]. rewrite -Hx; apply (proj2_sig (f i)). + destruct (m !! i) as [x|] eqn:Hx; rewrite /=; [apply (proj2_sig (f i))|]. move: (Hm12' i). rewrite Hx -!discrete_iff. rewrite !(symmetry_iff _ None) !equiv_None op_None; tauto. + destruct (m !! i) as [x|] eqn:Hx; simpl; [apply (proj2_sig (f i))|]. move: (Hm12' i). rewrite Hx -!discrete_iff. rewrite !(symmetry_iff _ None) !equiv_None op_None; tauto. Qed. Canonical Structure gmapR := CMRAT (gmap K A) gmap_cofe_mixin gmap_cmra_mixin. Global Instance gmap_cmra_discrete : CMRADiscrete A → CMRADiscrete gmapR. Proof. split; [apply _|]. intros m ? i. by apply: cmra_discrete_valid. Qed. Lemma gmap_ucmra_mixin : UCMRAMixin (gmap K A). Proof. split. - by intros i; rewrite lookup_empty. - by intros m i; rewrite /= lookup_op lookup_empty (left_id_L None _). - apply gmap_empty_timeless. - constructor=> i. by rewrite lookup_omap lookup_empty. Qed. Canonical Structure gmapUR := UCMRAT (gmap K A) gmap_cofe_mixin gmap_cmra_mixin gmap_ucmra_mixin. (** Internalized properties *) Lemma gmap_equivI {M} m1 m2 : m1 ≡ m2 ⊣⊢ (∀ i, m1 !! i ≡ m2 !! i : uPred M). Proof. by uPred.unseal. Qed. Lemma gmap_validI {M} m : ✓ m ⊣⊢ (∀ i, ✓ (m !! i) : uPred M). Proof. by uPred.unseal. Qed. End cmra. Arguments gmapR _ {_ _} _. Arguments gmapUR _ {_ _} _. Section properties. Context `{Countable K} {A : cmraT}. Implicit Types m : gmap K A. Implicit Types i : K. Implicit Types x y : A. Lemma lookup_opM m1 mm2 i : (m1 ⋅? mm2) !! i = m1 !! i ⋅ (mm2 ≫= (.!! i)). Proof. destruct mm2; by rewrite /= ?lookup_op ?right_id_L. Qed. Lemma lookup_validN_Some n m i x : ✓{n} m → m !! i ≡{n}≡ Some x → ✓{n} x. Proof. by move=> /(_ i) Hm Hi; move:Hm; rewrite Hi. Qed. Lemma lookup_valid_Some m i x : ✓ m → m !! i ≡ Some x → ✓ x. Proof. move=> Hm Hi. move:(Hm i). by rewrite Hi. Qed. Lemma insert_validN n m i x : ✓{n} x → ✓{n} m → ✓{n} <[i:=x]>m. Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_eq. Qed. Lemma insert_valid m i x : ✓ x → ✓ m → ✓ <[i:=x]>m. Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_eq. Qed. Lemma singleton_validN n i x : ✓{n} ({[ i := x ]} : gmap K A) ↔ ✓{n} x. Proof. split; [|by intros; apply insert_validN, ucmra_unit_validN]. by move=>/(_ i); simplify_map_eq. Qed. Lemma singleton_valid i x : ✓ ({[ i := x ]} : gmap K A) ↔ ✓ x. Proof. rewrite !cmra_valid_validN. by setoid_rewrite singleton_validN. Qed. Lemma insert_singleton_op m i x : m !! i = None → <[i:=x]> m = {[ i := x ]} ⋅ m. Proof. intros Hi; apply map_eq=> j; destruct (decide (i = j)) as [->|]. - by rewrite lookup_op lookup_insert lookup_singleton Hi right_id_L. - by rewrite lookup_op lookup_insert_ne // lookup_singleton_ne // left_id_L. Qed. Lemma core_singleton (i : K) (x : A) cx : pcore x = Some cx → core ({[ i := x ]} : gmap K A) = {[ i := cx ]}. Proof. apply omap_singleton. Qed. Lemma core_singleton' (i : K) (x : A) cx : pcore x ≡ Some cx → core ({[ i := x ]} : gmap K A) ≡ {[ i := cx ]}. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by rewrite (core_singleton _ _ cx'). Qed. Lemma op_singleton (i : K) (x y : A) : {[ i := x ]} ⋅ {[ i := y ]} = ({[ i := x ⋅ y ]} : gmap K A). Proof. by apply (merge_singleton _ _ _ x y). Qed. Global Instance gmap_persistent m : (∀ x : A, cmra.Persistent x) → cmra.Persistent m. Proof. intros; apply persistent_total=> i. rewrite lookup_core. apply (persistent_core _). Qed. Global Instance gmap_singleton_persistent i (x : A) : cmra.Persistent x → cmra.Persistent {[ i := x ]}. Proof. intros. by apply persistent_total, core_singleton'. Qed. Lemma singleton_includedN n m i x : {[ i := x ]} ≼{n} m ↔ ∃ y, m !! i ≡{n}≡ Some y ∧ (x ≼{n} y ∨ x ≡{n}≡ y). Proof. split. - move=> [m' /(_ i)]; rewrite lookup_op lookup_singleton. case (m' !! i)=> [y|]=> Hm. + exists (x ⋅ y); eauto using cmra_includedN_l. + exists x; eauto. - intros (y&Hi&[[z ?]| ->]). + exists (<[i:=z]>m)=> j; destruct (decide (i = j)) as [->|]. * rewrite Hi lookup_op lookup_singleton lookup_insert. by constructor. * by rewrite lookup_op lookup_singleton_ne // lookup_insert_ne // left_id. + exists (delete i m)=> j; destruct (decide (i = j)) as [->|]. * by rewrite Hi lookup_op lookup_singleton lookup_delete. * by rewrite lookup_op lookup_singleton_ne // lookup_delete_ne // left_id. Qed. Lemma dom_op m1 m2 : dom (gset K) (m1 ⋅ m2) = dom _ m1 ∪ dom _ m2. Proof. apply elem_of_equiv_L=> i; rewrite elem_of_union !elem_of_dom. unfold is_Some; setoid_rewrite lookup_op. destruct (m1 !! i), (m2 !! i); naive_solver. Qed. Lemma insert_updateP (P : A → Prop) (Q : gmap K A → Prop) m i x : x ~~>: P → (∀ y, P y → Q (<[i:=y]>m)) → <[i:=x]>m ~~>: Q. Proof. intros Hx%option_updateP' HP; apply cmra_total_updateP=> n mf Hm. destruct (Hx n (Some (mf !! i))) as ([y|]&?&?); try done. { by generalize (Hm i); rewrite lookup_op; simplify_map_eq. } exists (<[i:=y]> m); split; first by auto. intros j; move: (Hm j)=>{Hm}; rewrite !lookup_op=>Hm. destruct (decide (i = j)); simplify_map_eq/=; auto. Qed. Lemma insert_updateP' (P : A → Prop) m i x : x ~~>: P → <[i:=x]>m ~~>: λ m', ∃ y, m' = <[i:=y]>m ∧ P y. Proof. eauto using insert_updateP. Qed. Lemma insert_update m i x y : x ~~> y → <[i:=x]>m ~~> <[i:=y]>m. Proof. rewrite !cmra_update_updateP; eauto using insert_updateP with subst. Qed. Lemma singleton_updateP (P : A → Prop) (Q : gmap K A → Prop) i x : x ~~>: P → (∀ y, P y → Q {[ i := y ]}) → {[ i := x ]} ~~>: Q. Proof. apply insert_updateP. Qed. Lemma singleton_updateP' (P : A → Prop) i x : x ~~>: P → {[ i := x ]} ~~>: λ m, ∃ y, m = {[ i := y ]} ∧ P y. Proof. apply insert_updateP'. Qed. Lemma singleton_update i (x y : A) : x ~~> y → {[ i := x ]} ~~> {[ i := y ]}. Proof. apply insert_update. Qed. Lemma delete_update m i : m ~~> delete i m. Proof. apply cmra_total_update=> n mf Hm j; destruct (decide (i = j)); subst. - move: (Hm j). rewrite !lookup_op lookup_delete left_id. apply cmra_validN_op_r. - move: (Hm j). by rewrite !lookup_op lookup_delete_ne. Qed. Section freshness. Context `{Infinite K}. Lemma alloc_updateP_strong (Q : gmap K A → Prop) (I : gset K) m x : ✓ x → (∀ i, m !! i = None → i ∉ I → Q (<[i:=x]>m)) → m ~~>: Q. Proof. intros ? HQ. apply cmra_total_updateP. intros n mf Hm. set (i := fresh (I ∪ dom (gset K) (m ⋅ mf))). assert (i ∉ I ∧ i ∉ dom (gset K) m ∧ i ∉ dom (gset K) mf) as [?[??]]. { rewrite -not_elem_of_union -dom_op -not_elem_of_union; apply is_fresh. } exists (<[i:=x]>m); split. { apply HQ; last done. by eapply not_elem_of_dom. } rewrite insert_singleton_op; first by eapply not_elem_of_dom. rewrite -assoc -insert_singleton_op; first by eapply (not_elem_of_dom (D:=gset K)); rewrite dom_op not_elem_of_union. by apply insert_validN; [apply cmra_valid_validN|]. Qed. Lemma alloc_updateP (Q : gmap K A → Prop) m x : ✓ x → (∀ i, m !! i = None → Q (<[i:=x]>m)) → m ~~>: Q. Proof. move=>??. eapply alloc_updateP_strong with (I:=∅); by eauto. Qed. Lemma alloc_updateP_strong' m x (I : gset K) : ✓ x → m ~~>: λ m', ∃ i, i ∉ I ∧ m' = <[i:=x]>m ∧ m !! i = None. Proof. eauto using alloc_updateP_strong. Qed. Lemma alloc_updateP' m x : ✓ x → m ~~>: λ m', ∃ i, m' = <[i:=x]>m ∧ m !! i = None. Proof. eauto using alloc_updateP. Qed. Lemma alloc_unit_singleton_updateP (P : A → Prop) (Q : gmap K A → Prop) u i : ✓ u → LeftId (≡) u (⋅) → u ~~>: P → (∀ y, P y → Q {[ i := y ]}) → ∅ ~~>: Q. Proof. intros ?? Hx HQ. apply cmra_total_updateP=> n gf Hg. destruct (Hx n (gf !! i)) as (y&?&Hy). { move:(Hg i). rewrite !left_id. case: (gf !! i)=>[x|]; rewrite /= ?left_id //. intros; by apply cmra_valid_validN. } exists {[ i := y ]}; split; first by auto. intros i'; destruct (decide (i' = i)) as [->|]. - rewrite lookup_op lookup_singleton. move:Hy; case: (gf !! i)=>[x|]; rewrite /= ?right_id //. - move:(Hg i'). by rewrite !lookup_op lookup_singleton_ne // !left_id. Qed. Lemma alloc_unit_singleton_updateP' (P: A → Prop) u i : ✓ u → LeftId (≡) u (⋅) → u ~~>: P → ∅ ~~>: λ m, ∃ y, m = {[ i := y ]} ∧ P y. Proof. eauto using alloc_unit_singleton_updateP. Qed. Lemma alloc_unit_singleton_update u i (y : A) : ✓ u → LeftId (≡) u (⋅) → u ~~> y → ∅ ~~> {[ i := y ]}. Proof. rewrite !cmra_update_updateP; eauto using alloc_unit_singleton_updateP with subst. Qed. End freshness. Lemma insert_local_update m i x y mf : x ~l~> y @ mf ≫= (.!! i) → <[i:=x]>m ~l~> <[i:=y]>m @ mf. Proof. intros [Hxy Hxy']; split. - intros n Hm j. move: (Hm j). destruct (decide (i = j)); subst. + rewrite !lookup_opM !lookup_insert !Some_op_opM. apply Hxy. + by rewrite !lookup_opM !lookup_insert_ne. - intros n mf' Hm Hm' j. move: (Hm j) (Hm' j). destruct (decide (i = j)); subst. + rewrite !lookup_opM !lookup_insert !Some_op_opM !inj_iff. apply Hxy'. + by rewrite !lookup_opM !lookup_insert_ne. Qed. Lemma singleton_local_update i x y mf : x ~l~> y @ mf ≫= (.!! i) → {[ i := x ]} ~l~> {[ i := y ]} @ mf. Proof. apply insert_local_update. Qed. Lemma alloc_singleton_local_update m i x mf : (m ⋅? mf) !! i = None → ✓ x → m ~l~> <[i:=x]> m @ mf. Proof. rewrite lookup_opM op_None=> -[Hi Hif] ?. rewrite insert_singleton_op // comm. apply alloc_local_update. intros n Hm j. move: (Hm j). destruct (decide (i = j)); subst. - intros _; rewrite !lookup_opM lookup_op !lookup_singleton Hif Hi. by apply cmra_valid_validN. - by rewrite !lookup_opM lookup_op !lookup_singleton_ne // right_id. Qed. Lemma alloc_unit_singleton_local_update i x mf : mf ≫= (.!! i) = None → ✓ x → ∅ ~l~> {[ i := x ]} @ mf. Proof. intros Hi; apply alloc_singleton_local_update. by rewrite lookup_opM Hi. Qed. Lemma delete_local_update m i x `{!Exclusive x} mf : m !! i = Some x → m ~l~> delete i m @ mf. Proof. intros Hx; split; [intros n; apply delete_update|]. intros n mf' Hm Hm' j. move: (Hm j) (Hm' j). destruct (decide (i = j)); subst. + rewrite !lookup_opM !lookup_delete Hx=> ? Hj. rewrite (exclusiveN_Some_l n x (mf ≫= lookup j)) //. by rewrite (exclusiveN_Some_l n x (mf' ≫= lookup j)) -?Hj. + by rewrite !lookup_opM !lookup_delete_ne. Qed. End properties. (** Functor *) Instance gmap_fmap_ne `{Countable K} {A B : ofeT} (f : A → B) n : Proper (dist n ==> dist n) f → Proper (dist n ==>dist n) (fmap (M:=gmap K) f). Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed. Instance gmap_fmap_cmra_monotone `{Countable K} {A B : cmraT} (f : A → B) `{!CMRAMonotone f} : CMRAMonotone (fmap f : gmap K A → gmap K B). Proof. split; try apply _. - by intros n m ? i; rewrite lookup_fmap; apply (validN_preserving _). - intros m1 m2; rewrite !lookup_included=> Hm i. by rewrite !lookup_fmap; apply: cmra_monotone. Qed. Definition gmapO_map `{Countable K} {A B} (f: A -n> B) : gmapO K A -n> gmapO K B := OfeMor (fmap f : gmapO K A → gmapO K B). Instance gmapO_map_ne `{Countable K} {A B} n : Proper (dist n ==> dist n) (@gmapO_map K _ _ A B). Proof. intros f g Hf m k; rewrite /= !lookup_fmap. destruct (_ !! k) eqn:?; simpl; constructor; apply Hf. Qed. Program Definition gmapOF K `{Countable K} (F : oFunctor) : oFunctor := {| oFunctor_car A _ B _ := gmapO K (oFunctor_car F A B); oFunctor_map A1 _ A2 _ B1 _ B2 _ fg := gmapO_map (oFunctor_map F fg) |}. Next Obligation. by intros K ?? F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply gmapO_map_ne, oFunctor_ne. Qed. Next Obligation. intros K ?? F A ? B ? x. rewrite /= -{2}(map_fmap_id x). apply map_fmap_equiv_ext=>y ??; apply oFunctor_id. Qed. Next Obligation. intros K ?? F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -map_fmap_compose. apply map_fmap_equiv_ext=>y ??; apply oFunctor_compose. Qed. Instance gmapOF_contractive K `{Countable K} F : oFunctorContractive F → oFunctorContractive (gmapOF K F). Proof. by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply gmapO_map_ne, oFunctor_contractive. Qed. Program Definition gmapURF K `{Countable K} (F : rFunctor) : urFunctor := {| urFunctor_car A _ B _ := gmapUR K (rFunctor_car F A B); urFunctor_map A1 _ A2 _ B1 _ B2 _ fg := gmapO_map (rFunctor_map F fg) |}. Next Obligation. by intros K ?? F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply gmapO_map_ne, rFunctor_ne. Qed. Next Obligation. intros K ?? F A ? B ? x. rewrite /= -{2}(map_fmap_id x). apply map_fmap_equiv_ext=>y ??; apply rFunctor_id. Qed. Next Obligation. intros K ?? F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -map_fmap_compose. apply map_fmap_equiv_ext=>y ??; apply rFunctor_compose. Qed. Instance gmapRF_contractive K `{Countable K} F : rFunctorContractive F → urFunctorContractive (gmapURF K F). Proof. by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply gmapO_map_ne, rFunctor_contractive. Qed.