From iris.program_logic Require Export model. From iris.algebra Require Import iprod gmap. From iris.proofmode Require Import classes. Import uPred. (** The class [inG Σ A] expresses that the CMRA [A] is in the list of functors [Σ]. This class is similar to the [subG] class, but written down in terms of individual CMRAs instead of (lists of) CMRA *functors*. This additional class is needed because Coq is otherwise unable to solve type class constraints due to higher-order unification problems. *) Class inG (Σ : gFunctors) (A : cmraT) := InG { inG_id : gid Σ; inG_prf : A = Σ inG_id (iPreProp Σ) }. Arguments inG_id {_ _} _. Lemma subG_inG Σ (F : gFunctor) : subG F Σ → inG Σ (F (iPreProp Σ)). Proof. move=> /(_ 0%fin) /= [j ->]. by exists j. Qed. (** * Definition of the connective [own] *) Definition iRes_singleton `{i : inG Σ A} (γ : gname) (a : A) : iResUR Σ := iprod_singleton (inG_id i) {[ γ := cmra_transport inG_prf a ]}. Instance: Params (@iRes_singleton) 4. Definition own_def `{inG Σ A} (γ : gname) (a : A) : iProp Σ := uPred_ownM (iRes_singleton γ a). Definition own_aux : { x | x = @own_def }. by eexists. Qed. Definition own {Σ A i} := proj1_sig own_aux Σ A i. Definition own_eq : @own = @own_def := proj2_sig own_aux. Instance: Params (@own) 4. Typeclasses Opaque own. (** * Properties about ghost ownership *) Section global. Context `{i : inG Σ A}. Implicit Types a : A. (** ** Properties of [iRes_singleton] *) Global Instance iRes_singleton_ne γ n : Proper (dist n ==> dist n) (@iRes_singleton Σ A _ γ). Proof. by intros a a' Ha; apply iprod_singleton_ne; rewrite Ha. Qed. Lemma iRes_singleton_op γ a1 a2 : iRes_singleton γ (a1 ⋅ a2) ≡ iRes_singleton γ a1 ⋅ iRes_singleton γ a2. Proof. by rewrite /iRes_singleton iprod_op_singleton op_singleton cmra_transport_op. Qed. (** ** Properties of [own] *) Global Instance own_ne γ n : Proper (dist n ==> dist n) (@own Σ A _ γ). Proof. rewrite !own_eq. solve_proper. Qed. Global Instance own_proper γ : Proper ((≡) ==> (⊣⊢)) (@own Σ A _ γ) := ne_proper _. Lemma own_op γ a1 a2 : own γ (a1 ⋅ a2) ⊣⊢ own γ a1 ★ own γ a2. Proof. by rewrite !own_eq /own_def -ownM_op iRes_singleton_op. Qed. Lemma own_mono γ a1 a2 : a2 ≼ a1 → own γ a1 ⊢ own γ a2. Proof. move=> [c ->]. rewrite own_op. eauto with I. Qed. Global Instance own_cmra_homomorphism : CMRAHomomorphism (own γ). Proof. split. apply _. apply own_op. Qed. Global Instance own_mono' γ : Proper (flip (≼) ==> (⊢)) (@own Σ A _ γ). Proof. intros a1 a2. apply own_mono. Qed. Lemma own_valid γ a : own γ a ⊢ ✓ a. Proof. rewrite !own_eq /own_def ownM_valid /iRes_singleton. rewrite iprod_validI (forall_elim (inG_id i)) iprod_lookup_singleton. rewrite gmap_validI (forall_elim γ) lookup_singleton option_validI. (* implicit arguments differ a bit *) by trans (✓ cmra_transport inG_prf a : iProp Σ)%I; last destruct inG_prf. Qed. Lemma own_valid_2 γ a1 a2 : own γ a1 ★ own γ a2 ⊢ ✓ (a1 ⋅ a2). Proof. by rewrite -own_op own_valid. Qed. Lemma own_valid_3 γ a1 a2 a3 : own γ a1 ★ own γ a2 ★ own γ a3 ⊢ ✓ (a1 ⋅ a2 ⋅ a3). Proof. by rewrite -!own_op assoc own_valid. Qed. Lemma own_valid_r γ a : own γ a ⊢ own γ a ★ ✓ a. Proof. apply: uPred.always_entails_r. apply own_valid. Qed. Lemma own_valid_l γ a : own γ a ⊢ ✓ a ★ own γ a. Proof. by rewrite comm -own_valid_r. Qed. Global Instance own_timeless γ a : Timeless a → TimelessP (own γ a). Proof. rewrite !own_eq /own_def; apply _. Qed. Global Instance own_core_persistent γ a : Persistent a → PersistentP (own γ a). Proof. rewrite !own_eq /own_def; apply _. Qed. (** ** Allocation *) (* TODO: This also holds if we just have ✓ a at the current step-idx, as Iris assertion. However, the map_updateP_alloc does not suffice to show this. *) Lemma own_alloc_strong a (G : gset gname) : ✓ a → True ==★ ∃ γ, ■ (γ ∉ G) ∧ own γ a. Proof. intros Ha. rewrite -(bupd_mono (∃ m, ■ (∃ γ, γ ∉ G ∧ m = iRes_singleton γ a) ∧ uPred_ownM m)%I). - rewrite ownM_empty. eapply bupd_ownM_updateP, (iprod_singleton_updateP_empty (inG_id i)); first (eapply alloc_updateP_strong', cmra_transport_valid, Ha); naive_solver. - apply exist_elim=>m; apply pure_elim_l=>-[γ [Hfresh ->]]. by rewrite !own_eq /own_def -(exist_intro γ) pure_equiv // left_id. Qed. Lemma own_alloc a : ✓ a → True ==★ ∃ γ, own γ a. Proof. intros Ha. rewrite (own_alloc_strong a ∅) //; []. apply bupd_mono, exist_mono=>?. eauto with I. Qed. (** ** Frame preserving updates *) Lemma own_updateP P γ a : a ~~>: P → own γ a ==★ ∃ a', ■ P a' ∧ own γ a'. Proof. intros Ha. rewrite !own_eq. rewrite -(bupd_mono (∃ m, ■ (∃ a', m = iRes_singleton γ a' ∧ P a') ∧ uPred_ownM m)%I). - eapply bupd_ownM_updateP, iprod_singleton_updateP; first by (eapply singleton_updateP', cmra_transport_updateP', Ha). naive_solver. - apply exist_elim=>m; apply pure_elim_l=>-[a' [-> HP]]. rewrite -(exist_intro a'). by apply and_intro; [apply pure_intro|]. Qed. Lemma own_update γ a a' : a ~~> a' → own γ a ==★ own γ a'. Proof. intros; rewrite (own_updateP (a' =)); last by apply cmra_update_updateP. by apply bupd_mono, exist_elim=> a''; apply pure_elim_l=> ->. Qed. Lemma own_update_2 γ a1 a2 a' : a1 ⋅ a2 ~~> a' → own γ a1 ★ own γ a2 ==★ own γ a'. Proof. intros. rewrite -own_op. by apply own_update. Qed. Lemma own_update_3 γ a1 a2 a3 a' : a1 ⋅ a2 ⋅ a3 ~~> a' → own γ a1 ★ own γ a2 ★ own γ a3 ==★ own γ a'. Proof. intros. rewrite -!own_op assoc. by apply own_update. Qed. End global. Arguments own_valid {_ _} [_] _ _. Arguments own_valid_2 {_ _} [_] _ _ _. Arguments own_valid_3 {_ _} [_] _ _ _ _. Arguments own_valid_l {_ _} [_] _ _. Arguments own_valid_r {_ _} [_] _ _. Arguments own_updateP {_ _} [_] _ _ _ _. Arguments own_update {_ _} [_] _ _ _ _. Arguments own_update_2 {_ _} [_] _ _ _ _ _. Arguments own_update_3 {_ _} [_] _ _ _ _ _ _. Lemma own_empty `{inG Σ (A:ucmraT)} γ : True ==★ own γ ∅. Proof. rewrite ownM_empty !own_eq /own_def. apply bupd_ownM_update, iprod_singleton_update_empty. apply (alloc_unit_singleton_update (cmra_transport inG_prf ∅)); last done. - apply cmra_transport_valid, ucmra_unit_valid. - intros x; destruct inG_prf. by rewrite left_id. Qed. (** Proofmode class instances *) Section proofmode_classes. Context `{inG Σ A}. Implicit Types a b : A. Global Instance into_and_own p γ a b1 b2 : IntoOp a b1 b2 → IntoAnd p (own γ a) (own γ b1) (own γ b2). Proof. intros. apply mk_into_and_sep. by rewrite (into_op a) own_op. Qed. Global Instance from_sep_own γ a b1 b2 : FromOp a b1 b2 → FromSep (own γ a) (own γ b1) (own γ b2). Proof. intros. by rewrite /FromSep -own_op from_op. Qed. End proofmode_classes.