coq_tactics.v

From iris.algebra Require Export upred.
From iris.algebra Require Import upred_big_op upred_tactics.
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 env_spatial_is_nil {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 env_spatial_is_nil_persistent Δ :
  env_spatial_is_nil Δ = true → PersistentP Δ.
Proof. intros; destruct Δ as [? []]; simplify_eq/=; apply _. Qed.
Hint Immediate env_spatial_is_nil_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 HΔ; 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 :
  env_spatial_is_nil 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.

Lemma tac_timeless Δ Δ' i p P P' Q :
  IsExceptLast Q →
  envs_lookup i Δ = Some (p, P) → IntoExceptLast P P' →
  envs_simple_replace i p (Esnoc Enil i P') Δ = Some Δ' →
  (Δ' ⊢ Q) → Δ ⊢ Q.
Proof.
  intros ???? HQ. rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id HQ -{2}(is_except_last Q).
  by rewrite (into_except_last P) -except_last_always_if except_last_frame_r wand_elim_r.
Qed.

(** * Always *)
Lemma tac_always_intro Δ Q : env_spatial_is_nil 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 :
  env_spatial_is_nil 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.
Global Instance to_assert_rvs P Q : IntoAssert P (|=r=> Q) (|=r=> P).
Proof. by rewrite /IntoAssert rvs_frame_r wand_elim_r rvs_trans. 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.

Lemma tac_pose_proof Δ Δ' j P Q :
  (True ⊢ P) →
  envs_app true (Esnoc Enil j P) Δ = Some Δ' →
  (Δ' ⊢ Q) → Δ ⊢ Q.
Proof.
  intros HP ? <-. rewrite envs_app_sound //; simpl.
  by rewrite right_id -HP 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_and_destruct Δ Δ' i p j1 j2 P P1 P2 Q :
  envs_lookup i Δ = Some (p, P) → IntoAnd 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. rewrite (into_and p P).
  by destruct p; rewrite /= ?right_id (comm _ P1) ?always_and_sep wand_elim_r.
Qed.

(* Using this tactic, one can destruct a (non-separating) conjunction in the
spatial context as long as one of the conjuncts is thrown away. It corresponds
to the principle of "external choice" in linear logic. *)
Lemma tac_and_destruct_choice Δ Δ' i p (lr : bool) j P P1 P2 Q :
  envs_lookup i Δ = Some (p, P) → IntoAnd true P P1 P2 →
  envs_simple_replace i p (Esnoc Enil j (if lr then P1 else P2)) Δ = Some Δ' →
  (Δ' ⊢ Q) → Δ ⊢ Q.
Proof.
  intros. rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id (into_and true P). destruct lr.
  - by rewrite and_elim_l wand_elim_r.
  - by rewrite and_elim_r 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.