From iris.algebra Require Export upred. From iris.algebra Require Import upred_big_op upred_tactics gmap. From iris.proofmode Require Export environments classes. From iris.prelude Require Import stringmap hlist. Import uPred. Local Notation "Γ !! j" := (env_lookup j Γ). Local Notation "x ← y ; z" := (match y with Some x => z | None => None end). Local Notation "' ( x1 , x2 ) ← y ; z" := (match y with Some (x1,x2) => z | None => None end). Local Notation "' ( x1 , x2 , x3 ) ← y ; z" := (match y with Some (x1,x2,x3) => z | None => None end). Record envs (M : ucmraT) := Envs { env_persistent : env (uPred M); env_spatial : env (uPred M) }. Add Printing Constructor envs. Arguments Envs {_} _ _. Arguments env_persistent {_} _. Arguments env_spatial {_} _. Record envs_wf {M} (Δ : envs M) := { env_persistent_valid : env_wf (env_persistent Δ); env_spatial_valid : env_wf (env_spatial Δ); envs_disjoint i : env_persistent Δ !! i = None ∨ env_spatial Δ !! i = None }. Coercion of_envs {M} (Δ : envs M) : uPred M := (■ envs_wf Δ ★ □ [∧] env_persistent Δ ★ [★] env_spatial Δ)%I. Instance: Params (@of_envs) 1. Record envs_Forall2 {M} (R : relation (uPred M)) (Δ1 Δ2 : envs M) : Prop := { env_persistent_Forall2 : env_Forall2 R (env_persistent Δ1) (env_persistent Δ2); env_spatial_Forall2 : env_Forall2 R (env_spatial Δ1) (env_spatial Δ2) }. Definition envs_dom {M} (Δ : envs M) : list string := env_dom (env_persistent Δ) ++ env_dom (env_spatial Δ). Definition envs_lookup {M} (i : string) (Δ : envs M) : option (bool * uPred M) := let (Γp,Γs) := Δ in match env_lookup i Γp with | Some P => Some (true, P) | None => P ← env_lookup i Γs; Some (false, P) end. Definition envs_delete {M} (i : string) (p : bool) (Δ : envs M) : envs M := let (Γp,Γs) := Δ in match p with | true => Envs (env_delete i Γp) Γs | false => Envs Γp (env_delete i Γs) end. Definition envs_lookup_delete {M} (i : string) (Δ : envs M) : option (bool * uPred M * envs M) := let (Γp,Γs) := Δ in match env_lookup_delete i Γp with | Some (P,Γp') => Some (true, P, Envs Γp' Γs) | None => '(P,Γs') ← env_lookup_delete i Γs; Some (false, P, Envs Γp Γs') end. Definition envs_app {M} (p : bool) (Γ : env (uPred M)) (Δ : envs M) : option (envs M) := let (Γp,Γs) := Δ in match p with | true => _ ← env_app Γ Γs; Γp' ← env_app Γ Γp; Some (Envs Γp' Γs) | false => _ ← env_app Γ Γp; Γs' ← env_app Γ Γs; Some (Envs Γp Γs') end. Definition envs_simple_replace {M} (i : string) (p : bool) (Γ : env (uPred M)) (Δ : envs M) : option (envs M) := let (Γp,Γs) := Δ in match p with | true => _ ← env_app Γ Γs; Γp' ← env_replace i Γ Γp; Some (Envs Γp' Γs) | false => _ ← env_app Γ Γp; Γs' ← env_replace i Γ Γs; Some (Envs Γp Γs') end. Definition envs_replace {M} (i : string) (p q : bool) (Γ : env (uPred M)) (Δ : envs M) : option (envs M) := if eqb p q then envs_simple_replace i p Γ Δ else envs_app q Γ (envs_delete i p Δ). (* if [lr = false] then [result = (hyps named js, remaining hyps)], if [lr = true] then [result = (remaining hyps, hyps named js)] *) Definition envs_split {M} (lr : bool) (js : list string) (Δ : envs M) : option (envs M * envs M) := let (Γp,Γs) := Δ in '(Γs1,Γs2) ← env_split js Γs; match lr with | false => Some (Envs Γp Γs1, Envs Γp Γs2) | true => Some (Envs Γp Γs2, Envs Γp Γs1) end. Definition envs_persistent {M} (Δ : envs M) := if env_spatial Δ is Enil then true else false. Definition envs_clear_spatial {M} (Δ : envs M) : envs M := Envs (env_persistent Δ) Enil. (* Coq versions of the tactics *) Section tactics. Context {M : ucmraT}. Implicit Types Γ : env (uPred M). Implicit Types Δ : envs M. Implicit Types P Q : uPred M. Lemma of_envs_def Δ : of_envs Δ = (■ envs_wf Δ ★ □ [∧] env_persistent Δ ★ [★] env_spatial Δ)%I. Proof. done. Qed. Lemma envs_lookup_delete_Some Δ Δ' i p P : envs_lookup_delete i Δ = Some (p,P,Δ') ↔ envs_lookup i Δ = Some (p,P) ∧ Δ' = envs_delete i p Δ. Proof. rewrite /envs_lookup /envs_delete /envs_lookup_delete. destruct Δ as [Γp Γs]; rewrite /= !env_lookup_delete_correct. destruct (Γp !! i), (Γs !! i); naive_solver. Qed. Lemma envs_lookup_sound Δ i p P : envs_lookup i Δ = Some (p,P) → Δ ⊢ □?p P ★ envs_delete i p Δ. Proof. rewrite /envs_lookup /envs_delete /of_envs=>?; apply pure_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=. - rewrite (env_lookup_perm Γp) //= always_and_sep always_sep. ecancel [□ [∧] _; □ P; [★] _]%I; apply pure_intro. destruct Hwf; constructor; naive_solver eauto using env_delete_wf, env_delete_fresh. - destruct (Γs !! i) eqn:?; simplify_eq/=. rewrite (env_lookup_perm Γs) //=. ecancel [□ [∧] _; P; [★] _]%I; apply pure_intro. destruct Hwf; constructor; naive_solver eauto using env_delete_wf, env_delete_fresh. Qed. Lemma envs_lookup_sound' Δ i p P : envs_lookup i Δ = Some (p,P) → Δ ⊢ P ★ envs_delete i p Δ. Proof. intros. rewrite envs_lookup_sound //. by rewrite always_if_elim. Qed. Lemma envs_lookup_persistent_sound Δ i P : envs_lookup i Δ = Some (true,P) → Δ ⊢ □ P ★ Δ. Proof. intros. apply: always_entails_l. by rewrite envs_lookup_sound // sep_elim_l. Qed. Lemma envs_lookup_split Δ i p P : envs_lookup i Δ = Some (p,P) → Δ ⊢ □?p P ★ (□?p P -★ Δ). Proof. rewrite /envs_lookup /of_envs=>?; apply pure_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=. - rewrite (env_lookup_perm Γp) //= always_and_sep always_sep. rewrite pure_equiv // left_id. cancel [□ P]%I. apply wand_intro_l. solve_sep_entails. - destruct (Γs !! i) eqn:?; simplify_eq/=. rewrite (env_lookup_perm Γs) //=. rewrite pure_equiv // left_id. cancel [P]. apply wand_intro_l. solve_sep_entails. Qed. Lemma envs_lookup_delete_sound Δ Δ' i p P : envs_lookup_delete i Δ = Some (p,P,Δ') → Δ ⊢ □?p P ★ Δ'. Proof. intros [? ->]%envs_lookup_delete_Some. by apply envs_lookup_sound. Qed. Lemma envs_lookup_delete_sound' Δ Δ' i p P : envs_lookup_delete i Δ = Some (p,P,Δ') → Δ ⊢ P ★ Δ'. Proof. intros [? ->]%envs_lookup_delete_Some. by apply envs_lookup_sound'. Qed. Lemma envs_app_sound Δ Δ' p Γ : envs_app p Γ Δ = Some Δ' → Δ ⊢ □?p [★] Γ -★ Δ'. Proof. rewrite /of_envs /envs_app=> ?; apply pure_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], p; simplify_eq/=. - destruct (env_app Γ Γs) eqn:Happ, (env_app Γ Γp) as [Γp'|] eqn:?; simplify_eq/=. apply wand_intro_l, sep_intro_True_l; [apply pure_intro|]. + destruct Hwf; constructor; simpl; eauto using env_app_wf. intros j. apply (env_app_disjoint _ _ _ j) in Happ. naive_solver eauto using env_app_fresh. + rewrite (env_app_perm _ _ Γp') //. rewrite big_and_app always_and_sep always_sep (big_sep_and Γ). solve_sep_entails. - destruct (env_app Γ Γp) eqn:Happ, (env_app Γ Γs) as [Γs'|] eqn:?; simplify_eq/=. apply wand_intro_l, sep_intro_True_l; [apply pure_intro|]. + destruct Hwf; constructor; simpl; eauto using env_app_wf. intros j. apply (env_app_disjoint _ _ _ j) in Happ. naive_solver eauto using env_app_fresh. + rewrite (env_app_perm _ _ Γs') // big_sep_app. solve_sep_entails. Qed. Lemma envs_simple_replace_sound' Δ Δ' i p Γ : envs_simple_replace i p Γ Δ = Some Δ' → envs_delete i p Δ ⊢ □?p [★] Γ -★ Δ'. Proof. rewrite /envs_simple_replace /envs_delete /of_envs=> ?. apply pure_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], p; simplify_eq/=. - destruct (env_app Γ Γs) eqn:Happ, (env_replace i Γ Γp) as [Γp'|] eqn:?; simplify_eq/=. apply wand_intro_l, sep_intro_True_l; [apply pure_intro|]. + destruct Hwf; constructor; simpl; eauto using env_replace_wf. intros j. apply (env_app_disjoint _ _ _ j) in Happ. destruct (decide (i = j)); try naive_solver eauto using env_replace_fresh. + rewrite (env_replace_perm _ _ Γp') //. rewrite big_and_app always_and_sep always_sep (big_sep_and Γ). solve_sep_entails. - destruct (env_app Γ Γp) eqn:Happ, (env_replace i Γ Γs) as [Γs'|] eqn:?; simplify_eq/=. apply wand_intro_l, sep_intro_True_l; [apply pure_intro|]. + destruct Hwf; constructor; simpl; eauto using env_replace_wf. intros j. apply (env_app_disjoint _ _ _ j) in Happ. destruct (decide (i = j)); try naive_solver eauto using env_replace_fresh. + rewrite (env_replace_perm _ _ Γs') // big_sep_app. solve_sep_entails. Qed. Lemma envs_simple_replace_sound Δ Δ' i p P Γ : envs_lookup i Δ = Some (p,P) → envs_simple_replace i p Γ Δ = Some Δ' → Δ ⊢ □?p P ★ (□?p [★] Γ -★ Δ'). Proof. intros. by rewrite envs_lookup_sound// envs_simple_replace_sound'//. Qed. Lemma envs_replace_sound' Δ Δ' i p q Γ : envs_replace i p q Γ Δ = Some Δ' → envs_delete i p Δ ⊢ □?q [★] Γ -★ Δ'. Proof. rewrite /envs_replace; destruct (eqb _ _) eqn:Hpq. - apply eqb_prop in Hpq as ->. apply envs_simple_replace_sound'. - apply envs_app_sound. Qed. Lemma envs_replace_sound Δ Δ' i p q P Γ : envs_lookup i Δ = Some (p,P) → envs_replace i p q Γ Δ = Some Δ' → Δ ⊢ □?p P ★ (□?q [★] Γ -★ Δ'). Proof. intros. by rewrite envs_lookup_sound// envs_replace_sound'//. Qed. Lemma envs_split_sound Δ lr js Δ1 Δ2 : envs_split lr js Δ = Some (Δ1,Δ2) → Δ ⊢ Δ1 ★ Δ2. Proof. rewrite /envs_split /of_envs=> ?; apply pure_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], (env_split js _) as [[Γs1 Γs2]|] eqn:?; simplify_eq/=. rewrite (env_split_perm Γs) // big_sep_app {1}always_sep_dup'. destruct lr; simplify_eq/=; cancel [□ [∧] Γp; □ [∧] Γp; [★] Γs1; [★] Γs2]%I; destruct Hwf; apply sep_intro_True_l; apply pure_intro; constructor; naive_solver eauto using env_split_wf_1, env_split_wf_2, env_split_fresh_1, env_split_fresh_2. Qed. Lemma envs_clear_spatial_sound Δ : Δ ⊢ envs_clear_spatial Δ ★ [★] env_spatial Δ. Proof. rewrite /of_envs /envs_clear_spatial /=; apply pure_elim_sep_l=> Hwf. rewrite right_id -assoc; apply sep_intro_True_l; [apply pure_intro|done]. destruct Hwf; constructor; simpl; auto using Enil_wf. Qed. Lemma env_fold_wand Γ Q : env_fold uPred_wand Q Γ ⊣⊢ ([★] Γ -★ Q). Proof. revert Q; induction Γ as [|Γ IH i P]=> Q /=; [by rewrite wand_True|]. by rewrite IH wand_curry (comm uPred_sep). Qed. Lemma envs_persistent_persistent Δ : envs_persistent Δ = true → PersistentP Δ. Proof. intros; destruct Δ as [? []]; simplify_eq/=; apply _. Qed. Hint Immediate envs_persistent_persistent : typeclass_instances. Global Instance envs_Forall2_refl (R : relation (uPred M)) : Reflexive R → Reflexive (envs_Forall2 R). Proof. by constructor. Qed. Global Instance envs_Forall2_sym (R : relation (uPred M)) : Symmetric R → Symmetric (envs_Forall2 R). Proof. intros ??? [??]; by constructor. Qed. Global Instance envs_Forall2_trans (R : relation (uPred M)) : Transitive R → Transitive (envs_Forall2 R). Proof. intros ??? [??] [??] [??]; constructor; etrans; eauto. Qed. Global Instance envs_Forall2_antisymm (R R' : relation (uPred M)) : AntiSymm R R' → AntiSymm (envs_Forall2 R) (envs_Forall2 R'). Proof. intros ??? [??] [??]; constructor; by eapply (anti_symm _). Qed. Lemma envs_Forall2_impl (R R' : relation (uPred M)) Δ1 Δ2 : envs_Forall2 R Δ1 Δ2 → (∀ P Q, R P Q → R' P Q) → envs_Forall2 R' Δ1 Δ2. Proof. intros [??] ?; constructor; eauto using env_Forall2_impl. Qed. Global Instance of_envs_mono : Proper (envs_Forall2 (⊢) ==> (⊢)) (@of_envs M). Proof. intros [Γp1 Γs1] [Γp2 Γs2] [Hp Hs]; unfold of_envs; simpl in *. apply pure_elim_sep_l=>Hwf. apply sep_intro_True_l. - destruct Hwf; apply pure_intro; constructor; naive_solver eauto using env_Forall2_wf, env_Forall2_fresh. - by repeat f_equiv. Qed. Global Instance of_envs_proper : Proper (envs_Forall2 (⊣⊢) ==> (⊣⊢)) (@of_envs M). Proof. intros Δ1 Δ2 ?; apply (anti_symm (⊢)); apply of_envs_mono; eapply envs_Forall2_impl; [| |symmetry|]; eauto using equiv_entails. Qed. Global Instance Envs_mono (R : relation (uPred M)) : Proper (env_Forall2 R ==> env_Forall2 R ==> envs_Forall2 R) (@Envs M). Proof. by constructor. Qed. (** * Adequacy *) Lemma tac_adequate P : (Envs Enil Enil ⊢ P) → True ⊢ P. Proof. intros <-. rewrite /of_envs /= always_pure !right_id. apply pure_intro; repeat constructor. Qed. (** * Basic rules *) Lemma tac_assumption Δ i p P Q : envs_lookup i Δ = Some (p,P) → FromAssumption p P Q → Δ ⊢ Q. Proof. intros. by rewrite envs_lookup_sound // sep_elim_l. Qed. Lemma tac_rename Δ Δ' i j p P Q : envs_lookup i Δ = Some (p,P) → envs_simple_replace i p (Esnoc Enil j P) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros. rewrite envs_simple_replace_sound //. destruct p; simpl; by rewrite right_id wand_elim_r. Qed. Lemma tac_clear Δ Δ' i p P Q : envs_lookup_delete i Δ = Some (p,P,Δ') → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros. by rewrite envs_lookup_delete_sound // sep_elim_r. Qed. Lemma tac_clear_spatial Δ Δ' Q : envs_clear_spatial Δ = Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros <- ?. by rewrite envs_clear_spatial_sound // sep_elim_l. Qed. (** * False *) Lemma tac_ex_falso Δ Q : (Δ ⊢ False) → Δ ⊢ Q. Proof. by rewrite -(False_elim Q). Qed. (** * Pure *) Lemma tac_pure_intro Δ Q (φ : Prop) : FromPure Q φ → φ → Δ ⊢ Q. Proof. intros ??. rewrite -(from_pure Q) //. apply True_intro. Qed. Lemma tac_pure Δ Δ' i p P φ Q : envs_lookup_delete i Δ = Some (p, P, Δ') → IntoPure P φ → (φ → Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros ?? HQ. rewrite envs_lookup_delete_sound' //; simpl. rewrite (into_pure P); by apply pure_elim_sep_l. Qed. Lemma tac_pure_revert Δ φ Q : (Δ ⊢ ■ φ → Q) → (φ → Δ ⊢ Q). Proof. intros HΔ ?. by rewrite HΔ pure_equiv // left_id. Qed. (** * Later *) Class IntoLaterEnv (Γ1 Γ2 : env (uPred M)) := into_later_env : env_Forall2 IntoLater Γ1 Γ2. Class IntoLaterEnvs (Δ1 Δ2 : envs M) := { into_later_persistent: IntoLaterEnv (env_persistent Δ1) (env_persistent Δ2); into_later_spatial: IntoLaterEnv (env_spatial Δ1) (env_spatial Δ2) }. Global Instance into_later_env_nil : IntoLaterEnv Enil Enil. Proof. constructor. Qed. Global Instance into_later_env_snoc Γ1 Γ2 i P Q : IntoLaterEnv Γ1 Γ2 → IntoLater P Q → IntoLaterEnv (Esnoc Γ1 i P) (Esnoc Γ2 i Q). Proof. by constructor. Qed. Global Instance into_later_envs Γp1 Γp2 Γs1 Γs2 : IntoLaterEnv Γp1 Γp2 → IntoLaterEnv Γs1 Γs2 → IntoLaterEnvs (Envs Γp1 Γs1) (Envs Γp2 Γs2). Proof. by split. Qed. Lemma into_later_env_sound Δ1 Δ2 : IntoLaterEnvs Δ1 Δ2 → Δ1 ⊢ ▷ Δ2. Proof. intros [Hp Hs]; rewrite /of_envs /= !later_sep -always_later. repeat apply sep_mono; try apply always_mono. - rewrite -later_intro; apply pure_mono; destruct 1; constructor; naive_solver eauto using env_Forall2_wf, env_Forall2_fresh. - induction Hp; rewrite /= ?later_and; auto using and_mono, later_intro. - induction Hs; rewrite /= ?later_sep; auto using sep_mono, later_intro. Qed. Lemma tac_next Δ Δ' Q Q' : IntoLaterEnvs Δ Δ' → FromLater Q Q' → (Δ' ⊢ Q') → Δ ⊢ Q. Proof. intros ?? HQ. by rewrite -(from_later Q) into_later_env_sound HQ. Qed. Lemma tac_löb Δ Δ' i Q : envs_persistent Δ = true → envs_app true (Esnoc Enil i (▷ Q)%I) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros ?? HQ. rewrite -(always_elim Q) -(löb (□ Q)) -always_later. apply impl_intro_l, (always_intro _ _). rewrite envs_app_sound //; simpl. by rewrite right_id always_and_sep_l' wand_elim_r HQ. Qed. (** * Always *) Lemma tac_always_intro Δ Q : envs_persistent Δ = true → (Δ ⊢ Q) → Δ ⊢ □ Q. Proof. intros. by apply: always_intro. Qed. Lemma tac_persistent Δ Δ' i p P P' Q : envs_lookup i Δ = Some (p, P) → IntoPersistentP P P' → envs_replace i p true (Esnoc Enil i P') Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros ??? <-. rewrite envs_replace_sound //; simpl. by rewrite right_id (into_persistentP P) always_if_always wand_elim_r. Qed. (** * Implication and wand *) Lemma tac_impl_intro Δ Δ' i P Q : envs_persistent Δ = true → envs_app false (Esnoc Enil i P) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ P → Q. Proof. intros ?? HQ. rewrite (persistentP Δ) envs_app_sound //; simpl. by rewrite right_id always_wand_impl always_elim HQ. Qed. Lemma tac_impl_intro_persistent Δ Δ' i P P' Q : IntoPersistentP P P' → envs_app true (Esnoc Enil i P') Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ P → Q. Proof. intros ?? HQ. rewrite envs_app_sound //; simpl. apply impl_intro_l. by rewrite right_id {1}(into_persistentP P) always_and_sep_l wand_elim_r. Qed. Lemma tac_impl_intro_pure Δ P φ Q : IntoPure P φ → (φ → Δ ⊢ Q) → Δ ⊢ P → Q. Proof. intros. by apply impl_intro_l; rewrite (into_pure P); apply pure_elim_l. Qed. Lemma tac_wand_intro Δ Δ' i P Q : envs_app false (Esnoc Enil i P) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ P -★ Q. Proof. intros ? HQ. rewrite envs_app_sound //; simpl. by rewrite right_id HQ. Qed. Lemma tac_wand_intro_persistent Δ Δ' i P P' Q : IntoPersistentP P P' → envs_app true (Esnoc Enil i P') Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ P -★ Q. Proof. intros. rewrite envs_app_sound //; simpl. rewrite right_id. by apply wand_mono. Qed. Lemma tac_wand_intro_pure Δ P φ Q : IntoPure P φ → (φ → Δ ⊢ Q) → Δ ⊢ P -★ Q. Proof. intros. by apply wand_intro_l; rewrite (into_pure P); apply pure_elim_sep_l. Qed. (* This is pretty much [tac_specialize_assert] with [js:=[j]] and [tac_exact], but it is doing some work to keep the order of hypotheses preserved. *) Lemma tac_specialize Δ Δ' Δ'' i p j q P1 P2 R Q : envs_lookup_delete i Δ = Some (p, P1, Δ') → envs_lookup j (if p then Δ else Δ') = Some (q, R) → IntoWand R P1 P2 → match p with | true => envs_simple_replace j q (Esnoc Enil j P2) Δ | false => envs_replace j q false (Esnoc Enil j P2) Δ' (* remove [i] and make [j] spatial *) end = Some Δ'' → (Δ'' ⊢ Q) → Δ ⊢ Q. Proof. intros [? ->]%envs_lookup_delete_Some ??? <-. destruct p. - rewrite envs_lookup_persistent_sound // envs_simple_replace_sound //; simpl. rewrite assoc (into_wand R) (always_elim_if q) -always_if_sep wand_elim_r. by rewrite right_id wand_elim_r. - rewrite envs_lookup_sound //; simpl. rewrite envs_lookup_sound // (envs_replace_sound' _ Δ'') //; simpl. by rewrite right_id assoc (into_wand R) always_if_elim wand_elim_r wand_elim_r. Qed. Class IntoAssert (P : uPred M) (Q : uPred M) (R : uPred M) := into_assert : R ★ (P -★ Q) ⊢ Q. Global Arguments into_assert _ _ _ {_}. Lemma into_assert_default P Q : IntoAssert P Q P. Proof. by rewrite /IntoAssert wand_elim_r. Qed. Lemma tac_specialize_assert Δ Δ' Δ1 Δ2' j q lr js R P1 P2 P1' Q : envs_lookup_delete j Δ = Some (q, R, Δ') → IntoWand R P1 P2 → IntoAssert P1 Q P1' → ('(Δ1,Δ2) ← envs_split lr js Δ'; Δ2' ← envs_app false (Esnoc Enil j P2) Δ2; Some (Δ1,Δ2')) = Some (Δ1,Δ2') → (* does not preserve position of [j] *) (Δ1 ⊢ P1') → (Δ2' ⊢ Q) → Δ ⊢ Q. Proof. intros [? ->]%envs_lookup_delete_Some ??? HP1 HQ. destruct (envs_split _ _ _) as [[? Δ2]|] eqn:?; simplify_eq/=; destruct (envs_app _ _ _) eqn:?; simplify_eq/=. rewrite envs_lookup_sound // envs_split_sound //. rewrite (envs_app_sound Δ2) //; simpl. rewrite right_id (into_wand R) HP1 assoc -(comm _ P1') -assoc. rewrite -(into_assert P1 Q); apply sep_mono_r, wand_intro_l. by rewrite always_if_elim assoc !wand_elim_r. Qed. Lemma tac_specialize_pure Δ Δ' j q R P1 P2 φ Q : envs_lookup j Δ = Some (q, R) → IntoWand R P1 P2 → FromPure P1 φ → envs_simple_replace j q (Esnoc Enil j P2) Δ = Some Δ' → φ → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros. rewrite envs_simple_replace_sound //; simpl. by rewrite right_id (into_wand R) -(from_pure P1) // wand_True wand_elim_r. Qed. Lemma tac_specialize_persistent Δ Δ' Δ'' j q P1 P2 R Q : envs_lookup_delete j Δ = Some (q, R, Δ') → IntoWand R P1 P2 → envs_simple_replace j q (Esnoc Enil j P2) Δ = Some Δ'' → (Δ' ⊢ P1) → (PersistentP P1 ∨ PersistentP P2) → (Δ'' ⊢ Q) → Δ ⊢ Q. Proof. intros [? ->]%envs_lookup_delete_Some ?? HP1 [?|?] <-. - rewrite envs_lookup_sound //. rewrite -(idemp uPred_and (envs_delete _ _ _)). rewrite {1}HP1 (persistentP P1) always_and_sep_l assoc. rewrite envs_simple_replace_sound' //; simpl. rewrite right_id (into_wand R) (always_elim_if q) -always_if_sep wand_elim_l. by rewrite wand_elim_r. - rewrite -(idemp uPred_and Δ) {1}envs_lookup_sound //; simpl; rewrite HP1. rewrite envs_simple_replace_sound //; simpl. rewrite (sep_elim_r _ (_ -★ _)) right_id (into_wand R) always_if_elim. by rewrite wand_elim_l always_and_sep_l -{1}(always_if_always q P2) wand_elim_r. Qed. Lemma tac_revert Δ Δ' i p P Q : envs_lookup_delete i Δ = Some (p,P,Δ') → (Δ' ⊢ if p then □ P → Q else P -★ Q) → Δ ⊢ Q. Proof. intros ? HQ. rewrite envs_lookup_delete_sound //; simpl. destruct p. - by rewrite HQ -always_and_sep_l impl_elim_r. - by rewrite HQ wand_elim_r. Qed. Lemma tac_revert_spatial Δ Q : (envs_clear_spatial Δ ⊢ env_fold uPred_wand Q (env_spatial Δ)) → Δ ⊢ Q. Proof. intros HΔ. by rewrite envs_clear_spatial_sound HΔ env_fold_wand wand_elim_l. Qed. Lemma tac_assert Δ Δ1 Δ2 Δ2' lr js j P Q R : IntoAssert P Q R → envs_split lr js Δ = Some (Δ1,Δ2) → envs_app false (Esnoc Enil j P) Δ2 = Some Δ2' → (Δ1 ⊢ R) → (Δ2' ⊢ Q) → Δ ⊢ Q. Proof. intros ??? HP HQ. rewrite envs_split_sound //. rewrite (envs_app_sound Δ2) //; simpl. by rewrite right_id HP HQ. Qed. Lemma tac_assert_persistent Δ Δ' j P Q : envs_app true (Esnoc Enil j P) Δ = Some Δ' → (Δ ⊢ P) → PersistentP P → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros ? HP ??. rewrite -(idemp uPred_and Δ) {1}HP envs_app_sound //; simpl. by rewrite right_id {1}(persistentP P) always_and_sep_l wand_elim_r. Qed. (** Whenever posing [lem : True ⊢ Q] as [H] we want it to appear as [H : Q] and not as [H : True -★ Q]. The class [IntoPosedProof] is used to strip off the [True]. Note that [to_posed_proof_True] is declared using a [Hint Extern] to make sure it is not used while posing [lem : ?P ⊢ Q] with [?P] an evar. *) Class IntoPosedProof (P1 P2 R : uPred M) := into_pose_proof : (P1 ⊢ P2) → True ⊢ R. Arguments into_pose_proof : clear implicits. Instance to_posed_proof_True P : IntoPosedProof True P P. Proof. by rewrite /IntoPosedProof. Qed. Global Instance to_posed_proof_wand P Q : IntoPosedProof P Q (P -★ Q). Proof. rewrite /IntoPosedProof. apply entails_wand. Qed. Lemma tac_pose_proof Δ Δ' j P1 P2 R Q : (P1 ⊢ P2) → IntoPosedProof P1 P2 R → envs_app true (Esnoc Enil j R) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros HP ?? <-. rewrite envs_app_sound //; simpl. by rewrite right_id -(into_pose_proof P1 P2 R) // always_pure wand_True. Qed. Lemma tac_pose_proof_hyp Δ Δ' Δ'' i p j P Q : envs_lookup_delete i Δ = Some (p, P, Δ') → envs_app p (Esnoc Enil j P) (if p then Δ else Δ') = Some Δ'' → (Δ'' ⊢ Q) → Δ ⊢ Q. Proof. intros [? ->]%envs_lookup_delete_Some ? <-. destruct p. - rewrite envs_lookup_persistent_sound // envs_app_sound //; simpl. by rewrite right_id wand_elim_r. - rewrite envs_lookup_sound // envs_app_sound //; simpl. by rewrite right_id wand_elim_r. Qed. Lemma tac_apply Δ Δ' i p R P1 P2 : envs_lookup_delete i Δ = Some (p, R, Δ') → IntoWand R P1 P2 → (Δ' ⊢ P1) → Δ ⊢ P2. Proof. intros ?? HP1. rewrite envs_lookup_delete_sound' //. by rewrite (into_wand R) HP1 wand_elim_l. Qed. (** * Rewriting *) Lemma tac_rewrite Δ i p Pxy (lr : bool) Q : envs_lookup i Δ = Some (p, Pxy) → ∀ {A : cofeT} (x y : A) (Φ : A → uPred M), (Pxy ⊢ x ≡ y) → (Q ⊣⊢ Φ (if lr then y else x)) → (∀ n, Proper (dist n ==> dist n) Φ) → (Δ ⊢ Φ (if lr then x else y)) → Δ ⊢ Q. Proof. intros ? A x y ? HPxy -> ?; apply eq_rewrite; auto. rewrite {1}envs_lookup_sound' //; rewrite sep_elim_l HPxy. destruct lr; auto using eq_sym. Qed. Lemma tac_rewrite_in Δ i p Pxy j q P (lr : bool) Q : envs_lookup i Δ = Some (p, Pxy) → envs_lookup j Δ = Some (q, P) → ∀ {A : cofeT} Δ' x y (Φ : A → uPred M), (Pxy ⊢ x ≡ y) → (P ⊣⊢ Φ (if lr then y else x)) → (∀ n, Proper (dist n ==> dist n) Φ) → envs_simple_replace j q (Esnoc Enil j (Φ (if lr then x else y))) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros ?? A Δ' x y Φ HPxy HP ?? <-. rewrite -(idemp uPred_and Δ) {2}(envs_lookup_sound' _ i) //. rewrite sep_elim_l HPxy always_and_sep_r. rewrite (envs_simple_replace_sound _ _ j) //; simpl. rewrite HP right_id -assoc; apply wand_elim_r'. destruct lr. - apply (eq_rewrite x y (λ y, □?q Φ y -★ Δ')%I); eauto with I. solve_proper. - apply (eq_rewrite y x (λ y, □?q Φ y -★ Δ')%I); eauto using eq_sym with I. solve_proper. Qed. (** * Conjunction splitting *) Lemma tac_and_split Δ P Q1 Q2 : FromAnd P Q1 Q2 → (Δ ⊢ Q1) → (Δ ⊢ Q2) → Δ ⊢ P. Proof. intros. rewrite -(from_and P). by apply and_intro. Qed. (** * Separating conjunction splitting *) Lemma tac_sep_split Δ Δ1 Δ2 lr js P Q1 Q2 : FromSep P Q1 Q2 → envs_split lr js Δ = Some (Δ1,Δ2) → (Δ1 ⊢ Q1) → (Δ2 ⊢ Q2) → Δ ⊢ P. Proof. intros. rewrite envs_split_sound // -(from_sep P). by apply sep_mono. Qed. (** * Combining *) Lemma tac_combine Δ1 Δ2 Δ3 Δ4 i1 p P1 i2 q P2 j P Q : envs_lookup_delete i1 Δ1 = Some (p,P1,Δ2) → envs_lookup_delete i2 (if p then Δ1 else Δ2) = Some (q,P2,Δ3) → FromSep P P1 P2 → envs_app (p && q) (Esnoc Enil j P) (if q then (if p then Δ1 else Δ2) else Δ3) = Some Δ4 → (Δ4 ⊢ Q) → Δ1 ⊢ Q. Proof. intros [? ->]%envs_lookup_delete_Some [? ->]%envs_lookup_delete_Some ?? <-. destruct p. - rewrite envs_lookup_persistent_sound //. destruct q. + rewrite envs_lookup_persistent_sound // envs_app_sound //; simpl. by rewrite right_id assoc -always_sep (from_sep P) wand_elim_r. + rewrite envs_lookup_sound // envs_app_sound //; simpl. by rewrite right_id assoc always_elim (from_sep P) wand_elim_r. - rewrite envs_lookup_sound //; simpl. destruct q. + rewrite envs_lookup_persistent_sound // envs_app_sound //; simpl. by rewrite right_id assoc always_elim (from_sep P) wand_elim_r. + rewrite envs_lookup_sound // envs_app_sound //; simpl. by rewrite right_id assoc (from_sep P) wand_elim_r. Qed. (** * Conjunction/separating conjunction elimination *) Lemma tac_sep_destruct Δ Δ' i p j1 j2 P P1 P2 Q : envs_lookup i Δ = Some (p, P) → IntoSep p P P1 P2 → envs_simple_replace i p (Esnoc (Esnoc Enil j1 P1) j2 P2) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros. rewrite envs_simple_replace_sound //; simpl. by rewrite (into_sep p P) right_id (comm uPred_sep P1) wand_elim_r. Qed. (** * Framing *) Lemma tac_frame Δ Δ' i p R P Q : envs_lookup_delete i Δ = Some (p, R, Δ') → Frame R P Q → ((if p then Δ else Δ') ⊢ Q) → Δ ⊢ P. Proof. intros [? ->]%envs_lookup_delete_Some ? HQ. destruct p. - by rewrite envs_lookup_persistent_sound // always_elim -(frame R P) HQ. - rewrite envs_lookup_sound //; simpl. by rewrite -(frame R P) HQ. Qed. (** * Disjunction *) Lemma tac_or_l Δ P Q1 Q2 : FromOr P Q1 Q2 → (Δ ⊢ Q1) → Δ ⊢ P. Proof. intros. rewrite -(from_or P). by apply or_intro_l'. Qed. Lemma tac_or_r Δ P Q1 Q2 : FromOr P Q1 Q2 → (Δ ⊢ Q2) → Δ ⊢ P. Proof. intros. rewrite -(from_or P). by apply or_intro_r'. Qed. Lemma tac_or_destruct Δ Δ1 Δ2 i p j1 j2 P P1 P2 Q : envs_lookup i Δ = Some (p, P) → IntoOr P P1 P2 → envs_simple_replace i p (Esnoc Enil j1 P1) Δ = Some Δ1 → envs_simple_replace i p (Esnoc Enil j2 P2) Δ = Some Δ2 → (Δ1 ⊢ Q) → (Δ2 ⊢ Q) → Δ ⊢ Q. Proof. intros ???? HP1 HP2. rewrite envs_lookup_sound //. rewrite (into_or P) always_if_or sep_or_r; apply or_elim. - rewrite (envs_simple_replace_sound' _ Δ1) //. by rewrite /= right_id wand_elim_r. - rewrite (envs_simple_replace_sound' _ Δ2) //. by rewrite /= right_id wand_elim_r. Qed. (** * Forall *) Lemma tac_forall_intro {A} Δ (Φ : A → uPred M) : (∀ a, Δ ⊢ Φ a) → Δ ⊢ ∀ a, Φ a. Proof. apply forall_intro. Qed. Class ForallSpecialize {As} (xs : hlist As) (P : uPred M) (Φ : himpl As (uPred M)) := forall_specialize : P ⊢ Φ xs. Arguments forall_specialize {_} _ _ _ {_}. Global Instance forall_specialize_nil P : ForallSpecialize hnil P P | 100. Proof. done. Qed. Global Instance forall_specialize_cons A As x xs Φ (Ψ : A → himpl As (uPred M)) : (∀ x, ForallSpecialize xs (Φ x) (Ψ x)) → ForallSpecialize (hcons x xs) (∀ x : A, Φ x) Ψ. Proof. rewrite /ForallSpecialize /= => <-. by rewrite (forall_elim x). Qed. Lemma tac_forall_specialize {As} Δ Δ' i p P (Φ : himpl As (uPred M)) Q xs : envs_lookup i Δ = Some (p, P) → ForallSpecialize xs P Φ → envs_simple_replace i p (Esnoc Enil i (Φ xs)) Δ = Some Δ' → (Δ' ⊢ Q) → Δ ⊢ Q. Proof. intros. rewrite envs_simple_replace_sound //; simpl. by rewrite right_id (forall_specialize _ P) wand_elim_r. Qed. Lemma tac_forall_revert {A} Δ (Φ : A → uPred M) : (Δ ⊢ ∀ a, Φ a) → ∀ a, Δ ⊢ Φ a. Proof. intros HΔ a. by rewrite HΔ (forall_elim a). Qed. (** * Existential *) Lemma tac_exist {A} Δ P (Φ : A → uPred M) : FromExist P Φ → (∃ a, Δ ⊢ Φ a) → Δ ⊢ P. Proof. intros ? [a ?]. rewrite -(from_exist P). eauto using exist_intro'. Qed. Lemma tac_exist_destruct {A} Δ i p j P (Φ : A → uPred M) Q : envs_lookup i Δ = Some (p, P) → IntoExist P Φ → (∀ a, ∃ Δ', envs_simple_replace i p (Esnoc Enil j (Φ a)) Δ = Some Δ' ∧ (Δ' ⊢ Q)) → Δ ⊢ Q. Proof. intros ?? HΦ. rewrite envs_lookup_sound //. rewrite (into_exist P) always_if_exist sep_exist_r. apply exist_elim=> a; destruct (HΦ a) as (Δ'&?&?). rewrite envs_simple_replace_sound' //; simpl. by rewrite right_id wand_elim_r. Qed. End tactics. Hint Extern 0 (IntoPosedProof True _ _) => class_apply @to_posed_proof_True : typeclass_instances.