From algebra Require Export fin_maps agree excl functor. From algebra Require Import upred. From program_logic Require Export language. Record res (Λ : language) (Σ : iFunctor) (A : cofeT) := Res { wld : mapRA positive (agreeRA A); pst : exclRA (istateC Λ); gst : optionRA (Σ A); }. Add Printing Constructor res. Arguments Res {_ _ _} _ _ _. Arguments wld {_ _ _} _. Arguments pst {_ _ _} _. Arguments gst {_ _ _} _. Instance: Params (@Res) 3. Instance: Params (@wld) 3. Instance: Params (@pst) 3. Instance: Params (@gst) 3. Section res. Context {Λ : language} {Σ : iFunctor} {A : cofeT}. Implicit Types r : res Λ Σ A. Inductive res_equiv' (r1 r2 : res Λ Σ A) := Res_equiv : wld r1 ≡ wld r2 → pst r1 ≡ pst r2 → gst r1 ≡ gst r2 → res_equiv' r1 r2. Instance res_equiv : Equiv (res Λ Σ A) := res_equiv'. Inductive res_dist' (n : nat) (r1 r2 : res Λ Σ A) := Res_dist : wld r1 ≡{n}≡ wld r2 → pst r1 ≡{n}≡ pst r2 → gst r1 ≡{n}≡ gst r2 → res_dist' n r1 r2. Instance res_dist : Dist (res Λ Σ A) := res_dist'. Global Instance Res_ne n : Proper (dist n ==> dist n ==> dist n ==> dist n) (@Res Λ Σ A). Proof. done. Qed. Global Instance Res_proper : Proper ((≡) ==> (≡) ==> (≡) ==> (≡)) (@Res Λ Σ A). Proof. done. Qed. Global Instance wld_ne n : Proper (dist n ==> dist n) (@wld Λ Σ A). Proof. by destruct 1. Qed. Global Instance wld_proper : Proper ((≡) ==> (≡)) (@wld Λ Σ A). Proof. by destruct 1. Qed. Global Instance pst_ne n : Proper (dist n ==> dist n) (@pst Λ Σ A). Proof. by destruct 1. Qed. Global Instance pst_ne' n : Proper (dist n ==> (≡)) (@pst Λ Σ A). Proof. destruct 1; apply: timeless; apply dist_le with n; auto with lia. Qed. Global Instance pst_proper : Proper ((≡) ==> (=)) (@pst Λ Σ A). Proof. by destruct 1; unfold_leibniz. Qed. Global Instance gst_ne n : Proper (dist n ==> dist n) (@gst Λ Σ A). Proof. by destruct 1. Qed. Global Instance gst_proper : Proper ((≡) ==> (≡)) (@gst Λ Σ A). Proof. by destruct 1. Qed. Instance res_compl : Compl (res Λ Σ A) := λ c, Res (compl (chain_map wld c)) (compl (chain_map pst c)) (compl (chain_map gst c)). Definition res_cofe_mixin : CofeMixin (res Λ Σ A). Proof. split. - intros w1 w2; split. + by destruct 1; constructor; apply equiv_dist. + by intros Hw; constructor; apply equiv_dist=>n; destruct (Hw n). - intros n; split. + done. + by destruct 1; constructor. + do 2 destruct 1; constructor; etrans; eauto. - by destruct 1; constructor; apply dist_S. - intros n c; constructor. + apply (conv_compl n (chain_map wld c)). + apply (conv_compl n (chain_map pst c)). + apply (conv_compl n (chain_map gst c)). Qed. Canonical Structure resC : cofeT := CofeT res_cofe_mixin. Global Instance res_timeless r : Timeless (wld r) → Timeless (gst r) → Timeless r. Proof. by destruct 3; constructor; try apply: timeless. Qed. Instance res_op : Op (res Λ Σ A) := λ r1 r2, Res (wld r1 ⋅ wld r2) (pst r1 ⋅ pst r2) (gst r1 ⋅ gst r2). Global Instance res_empty : Empty (res Λ Σ A) := Res ∅ ∅ ∅. Instance res_unit : Unit (res Λ Σ A) := λ r, Res (unit (wld r)) (unit (pst r)) (unit (gst r)). Instance res_validN : ValidN (res Λ Σ A) := λ n r, ✓{n} wld r ∧ ✓{n} pst r ∧ ✓{n} gst r. Instance res_minus : Minus (res Λ Σ A) := λ r1 r2, Res (wld r1 ⩪ wld r2) (pst r1 ⩪ pst r2) (gst r1 ⩪ gst r2). Lemma res_included (r1 r2 : res Λ Σ A) : r1 ≼ r2 ↔ wld r1 ≼ wld r2 ∧ pst r1 ≼ pst r2 ∧ gst r1 ≼ gst r2. Proof. split; [|by intros ([w ?]&[σ ?]&[m ?]); exists (Res w σ m)]. intros [r Hr]; split_and?; [exists (wld r)|exists (pst r)|exists (gst r)]; apply Hr. Qed. Lemma res_includedN (r1 r2 : res Λ Σ A) n : r1 ≼{n} r2 ↔ wld r1 ≼{n} wld r2 ∧ pst r1 ≼{n} pst r2 ∧ gst r1 ≼{n} gst r2. Proof. split; [|by intros ([w ?]&[σ ?]&[m ?]); exists (Res w σ m)]. intros [r Hr]; split_and?; [exists (wld r)|exists (pst r)|exists (gst r)]; apply Hr. Qed. Definition res_cmra_mixin : CMRAMixin (res Λ Σ A). Proof. split. - by intros n x [???] ? [???]; constructor; simpl in *; cofe_subst. - by intros n [???] ? [???]; constructor; simpl in *; cofe_subst. - by intros n [???] ? [???] (?&?&?); split_and!; simpl in *; cofe_subst. - by intros n [???] ? [???] [???] ? [???]; constructor; simpl in *; cofe_subst. - by intros n ? (?&?&?); split_and!; apply cmra_validN_S. - by intros ???; constructor; rewrite /= assoc. - by intros ??; constructor; rewrite /= comm. - by intros ?; constructor; rewrite /= cmra_unit_l. - by intros ?; constructor; rewrite /= cmra_unit_idemp. - intros n r1 r2; rewrite !res_includedN. by intros (?&?&?); split_and!; apply cmra_unit_preservingN. - intros n r1 r2 (?&?&?); split_and!; simpl in *; eapply cmra_validN_op_l; eauto. - intros n r1 r2; rewrite res_includedN; intros (?&?&?). by constructor; apply cmra_op_minus. - intros n r r1 r2 (?&?&?) [???]; simpl in *. destruct (cmra_extend n (wld r) (wld r1) (wld r2)) as ([w w']&?&?&?), (cmra_extend n (pst r) (pst r1) (pst r2)) as ([σ σ']&?&?&?), (cmra_extend n (gst r) (gst r1) (gst r2)) as ([m m']&?&?&?); auto. by exists (Res w σ m, Res w' σ' m'). Qed. Canonical Structure resRA : cmraT := CMRAT res_cofe_mixin res_cmra_mixin. Global Instance res_cmra_identity : CMRAIdentity resRA. Proof. split. - intros n; split_and!; apply cmra_empty_valid. - by split; rewrite /= left_id. - apply _. Qed. Definition update_pst (σ : state Λ) (r : res Λ Σ A) : res Λ Σ A := Res (wld r) (Excl σ) (gst r). Definition update_gst (m : Σ A) (r : res Λ Σ A) : res Λ Σ A := Res (wld r) (pst r) (Some m). Lemma wld_validN n r : ✓{n} r → ✓{n} wld r. Proof. by intros (?&?&?). Qed. Lemma gst_validN n r : ✓{n} r → ✓{n} gst r. Proof. by intros (?&?&?). Qed. Lemma Res_op w1 w2 σ1 σ2 m1 m2 : Res w1 σ1 m1 ⋅ Res w2 σ2 m2 = Res (w1 ⋅ w2) (σ1 ⋅ σ2) (m1 ⋅ m2). Proof. done. Qed. Lemma Res_unit w σ m : unit (Res w σ m) = Res (unit w) (unit σ) (unit m). Proof. done. Qed. Lemma lookup_wld_op_l n r1 r2 i P : ✓{n} (r1⋅r2) → wld r1 !! i ≡{n}≡ Some P → (wld r1 ⋅ wld r2) !! i ≡{n}≡ Some P. Proof. move=>/wld_validN /(_ i) Hval Hi1P; move: Hi1P Hval; rewrite lookup_op. destruct (wld r2 !! i) as [P'|] eqn:Hi; rewrite !Hi ?right_id // =>-> ?. by constructor; rewrite (agree_op_inv _ P P') // agree_idemp. Qed. Lemma lookup_wld_op_r n r1 r2 i P : ✓{n} (r1⋅r2) → wld r2 !! i ≡{n}≡ Some P → (wld r1 ⋅ wld r2) !! i ≡{n}≡ Some P. Proof. rewrite (comm _ r1) (comm _ (wld r1)); apply lookup_wld_op_l. Qed. Global Instance Res_timeless eσ m : Timeless m → Timeless (Res ∅ eσ m). Proof. by intros ? ? [???]; constructor; apply: timeless. Qed. (** Internalized properties *) Lemma res_equivI {M} r1 r2 : (r1 ≡ r2)%I ≡ (wld r1 ≡ wld r2 ∧ pst r1 ≡ pst r2 ∧ gst r1 ≡ gst r2: uPred M)%I. Proof. do 2 split. by destruct 1. by intros (?&?&?); constructor. Qed. Lemma res_validI {M} r : (✓ r)%I ≡ (✓ wld r ∧ ✓ pst r ∧ ✓ gst r : uPred M)%I. Proof. done. Qed. End res. Arguments resC : clear implicits. Arguments resRA : clear implicits. Definition res_map {Λ Σ A B} (f : A -n> B) (r : res Λ Σ A) : res Λ Σ B := Res (agree_map f <\$> wld r) (pst r) (ifunctor_map Σ f <\$> gst r). Instance res_map_ne Λ Σ (A B : cofeT) (f : A -n> B) : (∀ n, Proper (dist n ==> dist n) f) → ∀ n, Proper (dist n ==> dist n) (@res_map Λ Σ _ _ f). Proof. by intros Hf n [] ? [???]; constructor; simpl in *; cofe_subst. Qed. Lemma res_map_id {Λ Σ A} (r : res Λ Σ A) : res_map cid r ≡ r. Proof. constructor; simpl; [|done|]. - rewrite -{2}(map_fmap_id (wld r)); apply map_fmap_setoid_ext=> i y ? /=. by rewrite -{2}(agree_map_id y); apply agree_map_ext. - rewrite -{2}(option_fmap_id (gst r)); apply option_fmap_setoid_ext=> m /=. by rewrite -{2}(ifunctor_map_id Σ m); apply ifunctor_map_ext. Qed. Lemma res_map_compose {Λ Σ A B C} (f : A -n> B) (g : B -n> C) (r : res Λ Σ A) : res_map (g ◎ f) r ≡ res_map g (res_map f r). Proof. constructor; simpl; [|done|]. - rewrite -map_fmap_compose; apply map_fmap_setoid_ext=> i y _ /=. by rewrite -agree_map_compose; apply agree_map_ext. - rewrite -option_fmap_compose; apply option_fmap_setoid_ext=> m /=. by rewrite -ifunctor_map_compose; apply ifunctor_map_ext. Qed. Lemma res_map_ext {Λ Σ A B} (f g : A -n> B) (r : res Λ Σ A) : (∀ x, f x ≡ g x) → res_map f r ≡ res_map g r. Proof. intros Hfg; split; simpl; auto. - by apply map_fmap_setoid_ext=>i x ?; apply agree_map_ext. - by apply option_fmap_setoid_ext=>m; apply ifunctor_map_ext. Qed. Instance res_map_cmra_monotone {Λ Σ} {A B : cofeT} (f : A -n> B) : CMRAMonotone (@res_map Λ Σ _ _ f). Proof. split. - by intros n r1 r2; rewrite !res_includedN; intros (?&?&?); split_and!; simpl; try apply includedN_preserving. - by intros n r (?&?&?); split_and!; simpl; try apply validN_preserving. Qed. Definition resC_map {Λ Σ A B} (f : A -n> B) : resC Λ Σ A -n> resC Λ Σ B := CofeMor (res_map f : resRA Λ Σ A → resRA Λ Σ B). Instance resC_map_ne {Λ Σ A B} n : Proper (dist n ==> dist n) (@resC_map Λ Σ A B). Proof. intros f g Hfg r; split; simpl; auto. - by apply (mapC_map_ne _ (agreeC_map f) (agreeC_map g)), agreeC_map_ne. - by apply optionC_map_ne, ifunctor_map_ne. Qed. Program Definition resF {Λ Σ} : iFunctor := {| ifunctor_car := resRA Λ Σ; ifunctor_map A B := resC_map |}. Next Obligation. intros Λ Σ A x. by rewrite /= res_map_id. Qed. Next Obligation. intros Λ Σ A B C f g x. by rewrite /= res_map_compose. Qed.