From iris.algebra Require Export cmra. From iris.prelude Require Export gmap. From iris.algebra Require Import upred. Section cofe. Context `{Countable K} {A : cofeT}. Implicit Types m : gmap K A. Instance gmap_dist : Dist (gmap K A) := λ n m1 m2, ∀ i, m1 !! i ≡{n}≡ m2 !! i. Program Definition gmap_chain (c : chain (gmap K A)) (k : K) : chain (option A) := {| chain_car n := c n !! k |}. Next Obligation. by intros c k n i ?; apply (chain_cauchy c). Qed. Instance gmap_compl : Compl (gmap K A) := λ c, map_imap (λ i _, compl (gmap_chain c i)) (c 0). Definition gmap_cofe_mixin : CofeMixin (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. - intros n c k; rewrite /compl /gmap_compl lookup_imap. feed inversion (λ H, chain_cauchy c 0 n H k); simpl; auto with lia. by rewrite conv_compl /=; apply reflexive_eq. Qed. Canonical Structure gmapC : cofeT := CofeT (gmap K A) gmap_cofe_mixin. Global Instance gmap_discrete : Discrete A → Discrete gmapC. Proof. intros ? m m' ? i. by apply (timeless _). Qed. (* why doesn't this go automatic? *) Global Instance gmapC_leibniz: LeibnizEquiv A → LeibnizEquiv gmapC. 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. Instance gmap_empty_timeless : Timeless (∅ : 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 : Timeless m → Timeless (m !! i). Proof. intros ? [x|] Hx; [|by symmetry; apply: timeless]. assert (m ≡{0}≡ <[i:=x]> m) by (by symmetry in Hx; inversion Hx; cofe_subst; rewrite insert_id). by rewrite (timeless m (<[i:=x]>m)) // lookup_insert. Qed. Global Instance gmap_insert_timeless m i x : Timeless x → Timeless m → Timeless (<[i:=x]>m). Proof. intros ?? m' Hm j; destruct (decide (i = j)); simplify_map_eq. { by apply: timeless; rewrite -Hm lookup_insert. } by apply: timeless; rewrite -Hm lookup_insert_ne. Qed. Global Instance gmap_singleton_timeless i x : Timeless x → Timeless ({[ i := x ]} : gmap K A) := _. End cofe. Arguments gmapC _ {_ _} _. (* 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_core : Core (gmap K A) := fmap core. 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). 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_fmap. 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: cmra_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. Definition gmap_cmra_mixin : CMRAMixin (gmap K A). Proof. split. - by intros n m1 m2 m3 Hm i; rewrite !lookup_op (Hm i). - by intros n m1 m2 Hm i; rewrite !lookup_core (Hm i). - by intros n m1 m2 Hm ? i; 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. - by intros m1 m2 m3 i; rewrite !lookup_op assoc. - by intros m1 m2 i; rewrite !lookup_op comm. - by intros m i; rewrite lookup_op !lookup_core cmra_core_l. - by intros m i; rewrite !lookup_core cmra_core_idemp. - intros x y; rewrite !lookup_included; intros Hm i. by rewrite !lookup_core; apply cmra_core_preserving. - 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 !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))|]. pose proof (Hm12' i) as Hm12''; rewrite Hx in Hm12''. by symmetry; apply option_op_positive_dist_l with (m2 !! i). + destruct (m !! i) as [x|] eqn:Hx; simpl; [apply (proj2_sig (f i))|]. pose proof (Hm12' i) as Hm12''; rewrite Hx in Hm12''. by symmetry; apply option_op_positive_dist_r with (m1 !! i). Qed. Canonical Structure gmapR : cmraT := CMRAT (gmap K A) gmap_cofe_mixin gmap_cmra_mixin. Global Instance gmap_cmra_unit : CMRAUnit gmapR. 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. Qed. Global Instance gmap_cmra_discrete : CMRADiscrete A → CMRADiscrete gmapR. Proof. split; [apply _|]. intros m ? i. by apply: cmra_discrete_valid. Qed. (** 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 _ {_ _} _. Section properties. Context `{Countable K} {A : cmraT}. Implicit Types m : gmap K A. Implicit Types i : K. Implicit Types a : A. 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, cmra_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_opN n m i x : m !! i = None ∨ m !! i ≡{n}≡ Some (core x) → <[i:=x]> m ≡{n}≡ {[ i := x ]} ⋅ m. Proof. intros Hi j; destruct (decide (i = j)) as [->|]; [|by rewrite lookup_op lookup_insert_ne // lookup_singleton_ne // left_id]. rewrite lookup_op lookup_insert lookup_singleton. by destruct Hi as [->| ->]; constructor; rewrite ?cmra_core_r. Qed. Lemma insert_singleton_op m i x : m !! i = None ∨ m !! i ≡ Some (core x) → <[i:=x]> m ≡ {[ i := x ]} ⋅ m. Proof. rewrite !equiv_dist; naive_solver eauto using insert_singleton_opN. Qed. Lemma core_singleton (i : K) (x : A) : core ({[ i := x ]} : gmap K A) = {[ i := core x ]}. Proof. apply map_fmap_singleton. 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, Persistent x) → Persistent m. Proof. intros ? i. by rewrite lookup_core persistent. Qed. Global Instance gmap_singleton_persistent i (x : A) : Persistent x → Persistent {[ i := x ]}. Proof. intros. by rewrite /Persistent core_singleton persistent. Qed. Lemma singleton_includedN n m i x : {[ i := x ]} ≼{n} m ↔ ∃ y, m !! i ≡{n}≡ Some 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. + by exists x. - 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. Qed. Lemma dom_op m1 m2 : dom (gset K) (m1 ⋅ m2) ≡ dom _ m1 ∪ dom _ m2. Proof. apply elem_of_equiv; intros 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 n mf Hm. destruct (Hx n (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 singleton_updateP_empty `{Empty A, !CMRAUnit A} (P : A → Prop) (Q : gmap K A → Prop) i : ∅ ~~>: P → (∀ y, P y → Q {[ i := y ]}) → ∅ ~~>: Q. Proof. intros Hx HQ n gf Hg. destruct (Hx n (from_option id ∅ (gf !! i))) as (y&?&Hy). { move:(Hg i). rewrite !left_id. case _: (gf !! i); simpl; auto using cmra_unit_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); first done. by rewrite right_id. - move:(Hg i'). by rewrite !lookup_op lookup_singleton_ne // !left_id. Qed. Lemma singleton_updateP_empty' `{Empty A, !CMRAUnit A} (P: A → Prop) i : ∅ ~~>: P → ∅ ~~>: λ m, ∃ y, m = {[ i := y ]} ∧ P y. Proof. eauto using singleton_updateP_empty. Qed. Section freshness. Context `{Fresh K (gset K), !FreshSpec K (gset K)}. Lemma updateP_alloc_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 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. { by apply HQ; last done; apply not_elem_of_dom. } rewrite insert_singleton_opN; last by left; apply not_elem_of_dom. rewrite -assoc -insert_singleton_opN; last by left; apply not_elem_of_dom; rewrite dom_op not_elem_of_union. by apply insert_validN; [apply cmra_valid_validN|]. Qed. Lemma updateP_alloc (Q : gmap K A → Prop) m x : ✓ x → (∀ i, m !! i = None → Q (<[i:=x]>m)) → m ~~>: Q. Proof. move=>??. eapply updateP_alloc_strong with (I:=∅); by eauto. Qed. Lemma updateP_alloc_strong' m x (I : gset K) : ✓ x → m ~~>: λ m', ∃ i, i ∉ I ∧ m' = <[i:=x]>m ∧ m !! i = None. Proof. eauto using updateP_alloc_strong. Qed. Lemma updateP_alloc' m x : ✓ x → m ~~>: λ m', ∃ i, m' = <[i:=x]>m ∧ m !! i = None. Proof. eauto using updateP_alloc. Qed. End freshness. (* Allocation is a local update: Just use composition with a singleton map. *) (* Deallocation is *not* a local update. The trouble is that if we own {[ i ↦ x ]}, then the frame could always own "core x", and prevent deallocation. *) (* Applying a local update at a position we own is a local update. *) Global Instance gmap_alter_update `{!LocalUpdate Lv L} i : LocalUpdate (λ m, ∃ x, m !! i = Some x ∧ Lv x) (alter L i). Proof. split; first apply _. intros n m1 m2 (x&Hix&?) Hm j; destruct (decide (i = j)) as [->|]. - rewrite lookup_alter !lookup_op lookup_alter Hix /=. move: (Hm j); rewrite lookup_op Hix. case: (m2 !! j)=>[y|] //=; constructor. by apply (local_updateN L). - by rewrite lookup_op !lookup_alter_ne // lookup_op. Qed. End properties. (** Functor *) Instance gmap_fmap_ne `{Countable K} {A B : cofeT} (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: included_preserving. Qed. Definition gmapC_map `{Countable K} {A B} (f: A -n> B) : gmapC K A -n> gmapC K B := CofeMor (fmap f : gmapC K A → gmapC K B). Instance gmapC_map_ne `{Countable K} {A B} n : Proper (dist n ==> dist n) (@gmapC_map K _ _ A B). Proof. intros f g Hf m k; rewrite /= !lookup_fmap. destruct (_ !! k) eqn:?; simpl; constructor; apply Hf. Qed. Program Definition gmapCF K `{Countable K} (F : cFunctor) : cFunctor := {| cFunctor_car A B := gmapC K (cFunctor_car F A B); cFunctor_map A1 A2 B1 B2 fg := gmapC_map (cFunctor_map F fg) |}. Next Obligation. by intros K ?? F A1 A2 B1 B2 n f g Hfg; apply gmapC_map_ne, cFunctor_ne. Qed. Next Obligation. intros K ?? F A B x. rewrite /= -{2}(map_fmap_id x). apply map_fmap_setoid_ext=>y ??; apply cFunctor_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_setoid_ext=>y ??; apply cFunctor_compose. Qed. Instance gmapCF_contractive K `{Countable K} F : cFunctorContractive F → cFunctorContractive (gmapCF K F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply gmapC_map_ne, cFunctor_contractive. Qed. Program Definition gmapRF K `{Countable K} (F : rFunctor) : rFunctor := {| rFunctor_car A B := gmapR K (rFunctor_car F A B); rFunctor_map A1 A2 B1 B2 fg := gmapC_map (rFunctor_map F fg) |}. Next Obligation. by intros K ?? F A1 A2 B1 B2 n f g Hfg; apply gmapC_map_ne, rFunctor_ne. Qed. Next Obligation. intros K ?? F A B x. rewrite /= -{2}(map_fmap_id x). apply map_fmap_setoid_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_setoid_ext=>y ??; apply rFunctor_compose. Qed. Instance gmapRF_contractive K `{Countable K} F : rFunctorContractive F → rFunctorContractive (gmapRF K F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply gmapC_map_ne, rFunctor_contractive. Qed.