Library iris.proofmode.coq_tactics

From iris.bi Require Export bi.
From iris.bi Require Import tactics.
From iris.proofmode Require Export base environments classes modality_instances.
Set Default Proof Using "Type".
Import bi.
Import env_notations.

Local Open Scope lazy_bool_scope.

Section bi_tactics.
Context {PROP : bi}.
Implicit Types Γ : env PROP.
Implicit Types Δ : envs PROP.
Implicit Types P Q : PROP.

Starting and stopping the proof mode

Lemma tac_start P : envs_entails (Envs Enil Enil 1) P bi_emp_valid P.
Proof.
  rewrite envs_entails_eq !of_envs_eq /=.
  rewrite intuitionistically_True_emp left_id=><-.
  apply and_intro⇒ //. apply pure_intro; repeat constructor.
Qed.

Lemma tac_stop Δ P :
  (match env_intuitionistic Δ, env_spatial Δ with
   | Enil, Γsenv_to_prop Γs
   | Γp, Enil env_to_prop_and Γp
   | Γp, Γs env_to_prop_and Γp env_to_prop Γs
   end%I P)
  envs_entails Δ P.
Proof.
  rewrite envs_entails_eq !of_envs_eq. intros <-.
  rewrite and_elim_r -env_to_prop_and_sound -env_to_prop_sound.
  destruct (env_intuitionistic Δ), (env_spatial Δ);
    by rewrite /= ?intuitionistically_True_emp ?left_id ?right_id.
Qed.

Basic rules

Lemma tac_eval Δ Q Q' :
  ( (Q'':=Q'), Q'' Q)
  envs_entails Δ Q' envs_entails Δ Q.
Proof. by intros <-. Qed.

Lemma tac_eval_in Δ i p P P' Q :
  envs_lookup i Δ = Some (p, P)
  ( (P'':=P'), P P')
  match envs_simple_replace i p (Esnoc Enil i P') Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite envs_entails_eq /=. intros ? HP ?.
  rewrite envs_simple_replace_singleton_sound //; simpl.
  by rewrite HP wand_elim_r.
Qed.

Class AffineEnv (Γ : env PROP) := affine_env : Forall Affine Γ.
Global Instance affine_env_nil : AffineEnv Enil.
Proof. constructor. Qed.
Global Instance affine_env_snoc Γ i P :
  Affine P AffineEnv Γ AffineEnv (Esnoc Γ i P).
Proof. by constructor. Qed.

Global Instance affine_env_bi `(BiAffine PROP) Γ : AffineEnv Γ | 0.
Proof. induction Γ; apply _. Qed.

Instance affine_env_spatial Δ :
  AffineEnv (env_spatial Δ) Affine ([∗] env_spatial Δ).
Proof. intros H. induction H; simpl; apply _. Qed.

Lemma tac_emp_intro Δ : AffineEnv (env_spatial Δ) envs_entails Δ emp.
Proof. intros. by rewrite envs_entails_eq (affine (of_envs Δ)). Qed.

Lemma tac_assumption Δ i p P Q :
  envs_lookup i Δ = Some (p,P)
  FromAssumption p P Q
  (let Δ' := envs_delete true i p Δ in
   if env_spatial_is_nil Δ' then TCTrue
   else TCOr (Absorbing Q) (AffineEnv (env_spatial Δ')))
  envs_entails Δ Q.
Proof.
  intros ?? H. rewrite envs_entails_eq envs_lookup_sound //.
  simpl in ×. destruct (env_spatial_is_nil _) eqn:?.
  - by rewrite (env_spatial_is_nil_intuitionistically _) // sep_elim_l.
  - rewrite from_assumption. destruct H; by rewrite sep_elim_l.
Qed.

Lemma tac_rename Δ i j p P Q :
  envs_lookup i Δ = Some (p,P)
  match envs_simple_replace i p (Esnoc Enil j P) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite envs_entails_eq⇒ ??. rewrite envs_simple_replace_singleton_sound //.
  by rewrite wand_elim_r.
Qed.

Lemma tac_clear Δ i p P Q :
  envs_lookup i Δ = Some (p,P)
  (if p then TCTrue else TCOr (Affine P) (Absorbing Q))
  envs_entails (envs_delete true i p Δ) Q
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq⇒ ?? HQ. rewrite envs_lookup_sound //. rewrite HQ.
  by destruct p; rewrite /= sep_elim_r.
Qed.

False

Lemma tac_ex_falso Δ Q : envs_entails Δ False envs_entails Δ Q.
Proof. by rewrite envs_entails_eq -(False_elim Q). Qed.

Lemma tac_false_destruct Δ i p P Q :
  envs_lookup i Δ = Some (p,P)
  P = False%I
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq ⇒ ??. subst. rewrite envs_lookup_sound //; simpl.
  by rewrite intuitionistically_if_elim sep_elim_l False_elim.
Qed.

Pure

Lemma tac_pure_intro Δ Q φ a :
  FromPure a Q φ
  (if a then AffineEnv (env_spatial Δ) else TCTrue)
  φ
  envs_entails Δ Q.
Proof.
  intros ???. rewrite envs_entails_eq -(from_pure a Q). destruct a; simpl.
  - by rewrite (affine (of_envs Δ)) pure_True // affinely_True_emp affinely_emp.
  - by apply pure_intro.
Qed.

Lemma tac_pure Δ i p P φ Q :
  envs_lookup i Δ = Some (p, P)
  IntoPure P φ
  (if p then TCTrue else TCOr (Affine P) (Absorbing Q))
  (φ envs_entails (envs_delete true i p Δ) Q) envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq⇒ ?? HPQ HQ.
  rewrite envs_lookup_sound //; simpl. destruct p; simpl.
  - rewrite (into_pure P) -persistently_and_intuitionistically_sep_l persistently_pure.
    by apply pure_elim_l.
  - destruct HPQ.
    + rewrite -(affine_affinely P) (into_pure P) -persistent_and_affinely_sep_l.
      by apply pure_elim_l.
    + rewrite (into_pure P) -(persistent_absorbingly_affinely _ %I) absorbingly_sep_lr.
      rewrite -persistent_and_affinely_sep_l. apply pure_elim_l⇒ ?. by rewrite HQ.
Qed.

Lemma tac_pure_revert Δ φ Q : envs_entails Δ (φ Q) (φ envs_entails Δ Q).
Proof. rewrite envs_entails_eq. intros ?. by rewrite pure_True // left_id. Qed.

Intuitionistic

Lemma tac_intuitionistic Δ i p P P' Q :
  envs_lookup i Δ = Some (p, P)
  IntoPersistent p P P'
  (if p then TCTrue else TCOr (Affine P) (Absorbing Q))
  match envs_replace i p true (Esnoc Enil i P') Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_replace _ _ _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite envs_entails_eq=>?? HPQ HQ. rewrite envs_replace_singleton_sound //=.
  destruct p; simpl; rewrite /bi_intuitionistically.
  - by rewrite -(into_persistent true P) /= wand_elim_r.
  - destruct HPQ.
    + rewrite -(affine_affinely P) (_ : P = <pers>?false P)%I //
              (into_persistent _ P) wand_elim_r //.
    + rewrite (_ : P = <pers>?false P)%I // (into_persistent _ P).
      by rewrite -{1}absorbingly_intuitionistically_into_persistently
        absorbingly_sep_l wand_elim_r HQ.
Qed.

Implication and wand

Lemma tac_impl_intro Δ i P P' Q R :
  FromImpl R P Q
  (if env_spatial_is_nil Δ then TCTrue else Persistent P)
  FromAffinely P' P
  match envs_app false (Esnoc Enil i P') Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ R.
Proof.
  destruct (envs_app _ _ _) eqn:?; last done.
  rewrite /FromImpl envs_entails_eq ⇒ <- ???. destruct (env_spatial_is_nil Δ) eqn:?.
  - rewrite (env_spatial_is_nil_intuitionistically Δ) //; simpl. apply impl_intro_l.
    rewrite envs_app_singleton_sound //; simpl.
    rewrite -(from_affinely P' P) -affinely_and_lr.
    by rewrite persistently_and_intuitionistically_sep_r intuitionistically_elim wand_elim_r.
  - apply impl_intro_l. rewrite envs_app_singleton_sound //; simpl.
    by rewrite -(from_affinely P' P) persistent_and_affinely_sep_l_1 wand_elim_r.
Qed.
Lemma tac_impl_intro_intuitionistic Δ i P P' Q R :
  FromImpl R P Q
  IntoPersistent false P P'
  match envs_app true (Esnoc Enil i P') Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ R.
Proof.
  destruct (envs_app _ _ _) eqn:?; last done.
  rewrite /FromImpl envs_entails_eq ⇒ <- ??.
  rewrite envs_app_singleton_sound //=. apply impl_intro_l.
  rewrite (_ : P = <pers>?false P)%I // (into_persistent false P).
  by rewrite persistently_and_intuitionistically_sep_l wand_elim_r.
Qed.
Lemma tac_impl_intro_drop Δ P Q R :
  FromImpl R P Q envs_entails Δ Q envs_entails Δ R.
Proof.
  rewrite /FromImpl envs_entails_eq ⇒ <- ?. apply impl_intro_l. by rewrite and_elim_r.
Qed.

Lemma tac_wand_intro Δ i P Q R :
  FromWand R P Q
  match envs_app false (Esnoc Enil i P) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ R.
Proof.
  destruct (envs_app _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite /FromWand envs_entails_eq ⇒ <- HQ.
  rewrite envs_app_sound //; simpl. by rewrite right_id HQ.
Qed.

Lemma tac_wand_intro_intuitionistic Δ i P P' Q R :
  FromWand R P Q
  IntoPersistent false P P'
  TCOr (Affine P) (Absorbing Q)
  match envs_app true (Esnoc Enil i P') Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ R.
Proof.
  destruct (envs_app _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite /FromWand envs_entails_eq ⇒ <- ? HPQ HQ.
  rewrite envs_app_singleton_sound //=. apply wand_intro_l. destruct HPQ.
  - rewrite -(affine_affinely P) (_ : P = <pers>?false P)%I //
            (into_persistent _ P) wand_elim_r //.
  - rewrite (_ : P = <pers>?false P)%I // (into_persistent _ P).
    by rewrite -{1}absorbingly_intuitionistically_into_persistently
      absorbingly_sep_l wand_elim_r HQ.
Qed.
Lemma tac_wand_intro_drop Δ P Q R :
  FromWand R P Q
  TCOr (Affine P) (Absorbing Q)
  envs_entails Δ Q
  envs_entails Δ R.
Proof.
  rewrite envs_entails_eq /FromWand ⇒ <- HPQ →. apply wand_intro_l. by rewrite sep_elim_r.
Qed.

Lemma tac_specialize remove_intuitionistic Δ i p j q P1 P2 R Q :
  envs_lookup i Δ = Some (p, P1)
  let Δ' := envs_delete remove_intuitionistic i p Δ in
  envs_lookup j Δ' = Some (q, R)
  IntoWand q p R P1 P2
  match envs_replace j q (p &&& q) (Esnoc Enil j P2) Δ' with
  | Some Δ''envs_entails Δ'' Q
  | NoneFalse
  end envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq /IntoWand. intros ?? HR ?.
  destruct (envs_replace _ _ _ _ _) as [Δ''|] eqn:?; last done.
  rewrite (envs_lookup_sound' _ remove_intuitionistic) //.
  rewrite envs_replace_singleton_sound //. destruct p; simpl in ×.
  - rewrite -{1}intuitionistically_idemp -{1}intuitionistically_if_idemp.
    rewrite {1}(intuitionistically_intuitionistically_if q).
    by rewrite HR assoc intuitionistically_if_sep_2 !wand_elim_r.
  - by rewrite HR assoc !wand_elim_r.
Qed.

Lemma tac_specialize_assert Δ j q neg js R P1 P2 P1' Q :
  envs_lookup j Δ = Some (q, R)
  IntoWand q false R P1 P2 AddModal P1' P1 Q
  match
    ''(Δ1,Δ2) envs_split (if neg is true then Right else Left)
    js (envs_delete true j q Δ);
    Δ2' envs_app false (Esnoc Enil j P2) Δ2;
    Some (Δ1,Δ2')
  with
  | Some (Δ1,Δ2')
     
     envs_entails Δ1 P1' envs_entails Δ2' Q
  | NoneFalse
  end envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq. intros ??? HQ.
  destruct (_ ≫= _) as [[Δ1 Δ2']|] eqn:?; last done.
  destruct HQ as [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_singleton_sound Δ2) //; simpl.
  rewrite HP1 (into_wand q false) /= -(add_modal P1' P1 Q). cancel [P1'].
  apply wand_intro_l. by rewrite assoc !wand_elim_r.
Qed.

Lemma tac_unlock_emp Δ Q : envs_entails Δ Q envs_entails Δ (emp locked Q).
Proof. rewrite envs_entails_eq⇒ →. by rewrite -lock left_id. Qed.
Lemma tac_unlock_True Δ Q : envs_entails Δ Q envs_entails Δ (True locked Q).
Proof. rewrite envs_entails_eq⇒ →. by rewrite -lock -True_sep_2. Qed.
Lemma tac_unlock Δ Q : envs_entails Δ Q envs_entails Δ (locked Q).
Proof. by unlock. Qed.

Lemma tac_specialize_frame Δ j q R P1 P2 P1' Q Q' :
  envs_lookup j Δ = Some (q, R)
  IntoWand q false R P1 P2
  AddModal P1' P1 Q
  envs_entails (envs_delete true j q Δ) (P1' locked Q')
  Q' = (P2 -∗ Q)%I
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq. intros ??? HPQ →.
  rewrite envs_lookup_sound //. rewrite HPQ -lock.
  rewrite (into_wand q false) -{2}(add_modal P1' P1 Q). cancel [P1'].
  apply wand_intro_l. by rewrite assoc !wand_elim_r.
Qed.

Lemma tac_specialize_assert_pure Δ j q a R P1 P2 φ Q :
  envs_lookup j Δ = Some (q, R)
  IntoWand q true R P1 P2
  FromPure a P1 φ
  φ
  match envs_simple_replace j q (Esnoc Enil j P2) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:?; last done.
  rewrite envs_entails_eq⇒ ?????. rewrite envs_simple_replace_singleton_sound //=.
  rewrite -intuitionistically_if_idemp (into_wand q true) /=.
  rewrite -(from_pure a P1) pure_True //.
  rewrite -affinely_affinely_if affinely_True_emp affinely_emp.
  by rewrite intuitionistically_emp left_id wand_elim_r.
Qed.

Lemma tac_specialize_assert_intuitionistic Δ j q P1 P1' P2 R Q :
  envs_lookup j Δ = Some (q, R)
  IntoWand q true R P1 P2
  Persistent P1
  IntoAbsorbingly P1' P1
  envs_entails (envs_delete true j q Δ) P1'
  match envs_simple_replace j q (Esnoc Enil j P2) Δ with
  | Some Δ''envs_entails Δ'' Q
  | NoneFalse
  end envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq ⇒ ???? HP1 HQ.
  destruct (envs_simple_replace _ _ _ _) as [Δ''|] eqn:?; last done.
  rewrite -HQ envs_lookup_sound //.
  rewrite -(idemp bi_and (of_envs (envs_delete _ _ _ _))).
  rewrite {2}envs_simple_replace_singleton_sound' //; simpl.
  rewrite {1}HP1 (into_absorbingly P1' P1) (persistent_persistently_2 P1).
  rewrite absorbingly_elim_persistently persistently_and_intuitionistically_sep_l assoc.
  rewrite -intuitionistically_if_idemp -intuitionistically_idemp.
  rewrite (intuitionistically_intuitionistically_if q).
  by rewrite intuitionistically_if_sep_2 (into_wand q true) wand_elim_l wand_elim_r.
Qed.

Lemma tac_specialize_intuitionistic_helper Δ j q P R R' Q :
  envs_lookup j Δ = Some (q,P)
  (if q then TCTrue else BiAffine PROP)
  envs_entails Δ (<absorb> R)
  IntoPersistent false R R'
  match envs_replace j q true (Esnoc Enil j R') Δ with
  | Some Δ''envs_entails Δ'' Q
  | NoneFalse
  end envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq ⇒ ?? HR ??.
  destruct (envs_replace _ _ _ _ _) as [Δ'|] eqn:?; last done.
  rewrite -(idemp bi_and (of_envs Δ)) {1}HR.
  rewrite envs_replace_singleton_sound //; destruct q; simpl.
  - by rewrite (_ : R = <pers>?false R)%I // (into_persistent _ R)
      absorbingly_elim_persistently sep_elim_r
      persistently_and_intuitionistically_sep_l wand_elim_r.
  - by rewrite (absorbing_absorbingly R) (_ : R = <pers>?false R)%I //
      (into_persistent _ R) sep_elim_r
      persistently_and_intuitionistically_sep_l wand_elim_r.
Qed.

Lemma tac_specialize_intuitionistic_helper_done Δ i q P :
  envs_lookup i Δ = Some (q,P)
  envs_entails Δ (<absorb> P).
Proof.
  rewrite envs_entails_eq /bi_absorbingly⇒ /envs_lookup_sound⇒ →.
  rewrite intuitionistically_if_elim comm. f_equiv; auto using pure_intro.
Qed.

Lemma tac_revert Δ i Q :
  match envs_lookup_delete true i Δ with
  | Some (p,P,Δ')envs_entails Δ' ((if p then P else P)%I -∗ Q)
  | NoneFalse
  end
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eqHQ.
  destruct (envs_lookup_delete _ _ _) as [[[p P] Δ']|] eqn:?; last done.
  rewrite envs_lookup_delete_sound //=.
  rewrite HQ. destruct p; simpl; auto using wand_elim_r.
Qed.

Class IntoIH (φ : Prop) (Δ : envs PROP) (Q : PROP) :=
  into_ih : φ of_envs Δ Q.
Global Instance into_ih_entails Δ Q : IntoIH (envs_entails Δ Q) Δ Q.
Proof. by rewrite envs_entails_eq /IntoIH. Qed.
Global Instance into_ih_forall {A} (φ : A Prop) Δ Φ :
  ( x, IntoIH (φ x) Δ (Φ x)) IntoIH ( x, φ x) Δ ( x, Φ x)%I | 2.
Proof. rewrite /IntoIH ?. apply forall_introx. by rewrite ( x). Qed.
Global Instance into_ih_impl (φ ψ : Prop) Δ Q :
  IntoIH φ Δ Q IntoIH (ψ φ) Δ (ψ Q)%I | 1.
Proof. rewrite /IntoIH ?. apply impl_intro_l, pure_elim_l. auto. Qed.

Lemma tac_revert_ih Δ P Q {φ : Prop} (Hφ : φ) :
  IntoIH φ Δ P
  env_spatial_is_nil Δ = true
  envs_entails Δ (<pers> P Q)
  envs_entails Δ Q.
Proof.
  rewrite /IntoIH envs_entails_eq. intros HP ? HPQ.
  rewrite (env_spatial_is_nil_intuitionistically Δ) //.
  rewrite -(idemp bi_and ( (of_envs Δ))%I) {1}HP // HPQ.
  rewrite {1}intuitionistically_into_persistently_1 intuitionistically_elim impl_elim_r //.
Qed.

Lemma tac_assert Δ j P Q :
  match envs_app true (Esnoc Enil j (P -∗ P)%I) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end envs_entails Δ Q.
Proof.
  destruct (envs_app _ _ _) as [Δ'|] eqn:?; last done.
  rewrite envs_entails_eq⇒ ?. rewrite (envs_app_singleton_sound Δ) //; simpl.
  by rewrite -(entails_wand P) // intuitionistically_emp emp_wand.
Qed.

Definition IntoEmpValid (φ : Type) (P : PROP) := φ bi_emp_valid P.
These lemmas are Defined because the guardedness checker must see through them. See https://gitlab.mpi-sws.org/iris/iris/issues/274. For the same reason, their bodies use as little automation as possible.
Lemma into_emp_valid_here φ P : AsEmpValid φ P IntoEmpValid φ P.
Proof. by intros [??]. Defined.
Lemma into_emp_valid_impl (φ ψ : Type) P :
  φ IntoEmpValid ψ P IntoEmpValid (φ ψ) P.
Proof. rewrite /IntoEmpValidHφ Hi1 Hi2. apply Hi1, Hi2, Hφ. Defined.
Lemma into_emp_valid_forall {A} (φ : A Type) P x :
  IntoEmpValid (φ x) P IntoEmpValid ( x : A, φ x) P.
Proof. rewrite /IntoEmpValidHi1 Hi2. apply Hi1, Hi2. Defined.
Lemma into_emp_valid_proj φ P : IntoEmpValid φ P φ bi_emp_valid P.
Proof. intros HP. apply HP. Defined.

When called by the proof mode, the proof of P is produced by calling into_emp_valid_proj. That call must be transparent to the guardedness checker, per https://gitlab.mpi-sws.org/iris/iris/issues/274; hence, it must be done outside tac_pose_proof, so the latter can remain opaque.
Lemma tac_pose_proof Δ j P Q :
  P
  match envs_app true (Esnoc Enil j P) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_app _ _ _) as [Δ'|] eqn:?; last done.
  rewrite envs_entails_eqHP <-. rewrite envs_app_singleton_sound //=.
  by rewrite -HP /= intuitionistically_emp emp_wand.
Qed.

Lemma tac_pose_proof_hyp Δ i j Q :
  match envs_lookup_delete false i Δ with
  | NoneFalse
  | Some (p, P, Δ')
    match envs_app p (Esnoc Enil j P) Δ' with
    | NoneFalse
    | Some Δ''envs_entails Δ'' Q
    end
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_lookup_delete _ _ _) as [((p&P)&Δ')|] eqn:Hlookup; last done.
  destruct (envs_app _ _ _) as [Δ''|] eqn:?; last done.
  rewrite envs_entails_eq. rewrite envs_lookup_delete_Some in Hlookup ×.
  intros [? ->] <-.
  rewrite envs_lookup_sound' // envs_app_singleton_sound //=.
  by rewrite wand_elim_r.
Qed.

Lemma tac_apply Δ i p R P1 P2 :
  envs_lookup i Δ = Some (p, R)
  IntoWand p false R P1 P2
  envs_entails (envs_delete true i p Δ) P1 envs_entails Δ P2.
Proof.
  rewrite envs_entails_eq ⇒ ?? HP1. rewrite envs_lookup_sound //.
  by rewrite (into_wand p false) /= HP1 wand_elim_l.
Qed.

Conjunction splitting

Lemma tac_and_split Δ P Q1 Q2 :
  FromAnd P Q1 Q2 envs_entails Δ Q1 envs_entails Δ Q2 envs_entails Δ P.
Proof. rewrite envs_entails_eq. intros. rewrite -(from_and P). by apply and_intro. Qed.

Separating conjunction splitting

Lemma tac_sep_split Δ d js P Q1 Q2 :
  FromSep P Q1 Q2
  match envs_split d js Δ with
  | NoneFalse
  | Some (Δ1,Δ2)envs_entails Δ1 Q1 envs_entails Δ2 Q2
  end envs_entails Δ P.
Proof.
  destruct (envs_split _ _ _) as [(Δ1&Δ2)|] eqn:?; last done.
  rewrite envs_entails_eq=>? [HQ1 HQ2].
  rewrite envs_split_sound //. by rewrite HQ1 HQ2.
Qed.

Combining

Class FromSeps {PROP : bi} (P : PROP) (Qs : list PROP) :=
  from_seps : [∗] Qs P.
Arguments FromSeps {_} _%I _%I.
Arguments from_seps {_} _%I _%I {_}.

Global Instance from_seps_nil : @FromSeps PROP emp [].
Proof. by rewrite /FromSeps. Qed.
Global Instance from_seps_singleton P : FromSeps P [P] | 1.
Proof. by rewrite /FromSeps /= right_id. Qed.
Global Instance from_seps_cons P P' Q Qs :
  FromSeps P' Qs FromSep P Q P' FromSeps P (Q :: Qs) | 2.
Proof. by rewrite /FromSeps /FromSep /= ⇒ →. Qed.

Lemma tac_combine Δ1 Δ2 Δ3 js p Ps j P Q :
  envs_lookup_delete_list false js Δ1 = Some (p, Ps, Δ2)
  FromSeps P Ps
  envs_app p (Esnoc Enil j P) Δ2 = Some Δ3
  envs_entails Δ3 Q envs_entails Δ1 Q.
Proof.
  rewrite envs_entails_eq ⇒ ??? <-. rewrite envs_lookup_delete_list_sound //.
  rewrite from_seps. rewrite envs_app_singleton_sound //=.
  by rewrite wand_elim_r.
Qed.

Conjunction/separating conjunction elimination

Lemma tac_and_destruct Δ i p j1 j2 P P1 P2 Q :
  envs_lookup i Δ = Some (p, P)
  (if p then IntoAnd true P P1 P2 else IntoSep P P1 P2)
  match envs_simple_replace i p (Esnoc (Esnoc Enil j1 P1) j2 P2) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite envs_entails_eq. intros. rewrite envs_simple_replace_sound //=. destruct p.
  - by rewrite (into_and _ P) /= right_id -(comm _ P1) wand_elim_r.
  - by rewrite /= (into_sep P) right_id -(comm _ P1) wand_elim_r.
Qed.

Lemma tac_and_destruct_choice Δ i p d j P P1 P2 Q :
  envs_lookup i Δ = Some (p, P)
  (if p then IntoAnd p P P1 P2 : Type else
    TCOr (IntoAnd p P P1 P2) (TCAnd (IntoSep P P1 P2)
      (if d is Left then TCOr (Affine P2) (Absorbing Q)
       else TCOr (Affine P1) (Absorbing Q))))
  match envs_simple_replace i p (Esnoc Enil j (if d is Left then P1 else P2)) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end envs_entails Δ Q.
Proof.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite envs_entails_eq ⇒ ? HP HQ.
  rewrite envs_simple_replace_singleton_sound //=. destruct p.
  { rewrite (into_and _ P) HQ. destruct d; simpl.
    - by rewrite and_elim_l wand_elim_r.
    - by rewrite and_elim_r wand_elim_r. }
  destruct HP as [?|[??]].
  { rewrite (into_and _ P) HQ. destruct d; simpl.
    - by rewrite and_elim_l wand_elim_r.
    - by rewrite and_elim_r wand_elim_r. }
  rewrite (into_sep P) HQ. destruct d; simpl.
  - by rewrite (comm _ P1) -assoc wand_elim_r sep_elim_r.
  - by rewrite -assoc wand_elim_r sep_elim_r.
Qed.

Framing

Lemma tac_frame_pure Δ (φ : Prop) P Q :
  φ Frame true φ P Q envs_entails Δ Q envs_entails Δ P.
Proof.
  rewrite envs_entails_eq ⇒ ? Hframe →. rewrite -Hframe /=.
  rewrite -persistently_and_intuitionistically_sep_l persistently_pure.
  auto using and_intro, pure_intro.
Qed.

Lemma tac_frame Δ i p R P Q :
  envs_lookup i Δ = Some (p, R)
  Frame p R P Q
  envs_entails (envs_delete false i p Δ) Q envs_entails Δ P.
Proof.
  rewrite envs_entails_eq. intros ? Hframe HQ.
  rewrite (envs_lookup_sound' _ false) //. by rewrite -Hframe HQ.
Qed.

Disjunction

Lemma tac_or_l Δ P Q1 Q2 :
  FromOr P Q1 Q2 envs_entails Δ Q1 envs_entails Δ P.
Proof.
  rewrite envs_entails_eq⇒ ? →. rewrite -(from_or P). by apply or_intro_l'.
Qed.
Lemma tac_or_r Δ P Q1 Q2 :
  FromOr P Q1 Q2 envs_entails Δ Q2 envs_entails Δ P.
Proof.
  rewrite envs_entails_eq⇒ ? →. rewrite -(from_or P). by apply or_intro_r'.
Qed.

Lemma tac_or_destruct Δ i p j1 j2 P P1 P2 Q :
  envs_lookup i Δ = Some (p, P) IntoOr P P1 P2
  match envs_simple_replace i p (Esnoc Enil j1 P1) Δ,
        envs_simple_replace i p (Esnoc Enil j2 P2) Δ with
  | Some Δ1, Some Δ2envs_entails Δ1 Q envs_entails Δ2 Q
  | _, _False
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_simple_replace i p (Esnoc Enil j1 P1)) as [Δ1|] eqn:?; last done.
  destruct (envs_simple_replace i p (Esnoc Enil j2 P2)) as [Δ2|] eqn:?; last done.
  rewrite envs_entails_eq. intros ?? (HP1&HP2). rewrite envs_lookup_sound //.
  rewrite (into_or P) intuitionistically_if_or sep_or_r; apply or_elim.
  - rewrite (envs_simple_replace_singleton_sound' _ Δ1) //.
    by rewrite wand_elim_r.
  - rewrite (envs_simple_replace_singleton_sound' _ Δ2) //.
    by rewrite wand_elim_r.
Qed.

Forall

Lemma tac_forall_intro {A} Δ (Φ : A PROP) Q :
  FromForall Q Φ
  ( a, envs_entails Δ (Φ a))
  envs_entails Δ Q.
Proof. rewrite envs_entails_eq /FromForall⇒ <-. apply forall_intro. Qed.

Lemma tac_forall_specialize {A} Δ i p P (Φ : A PROP) Q :
  envs_lookup i Δ = Some (p, P) IntoForall P Φ
  ( x : A,
      match envs_simple_replace i p (Esnoc Enil i (Φ x)) Δ with
      | NoneFalse
      | Some Δ'envs_entails Δ' Q
      end)
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq. intros ?? (x&?).
  destruct (envs_simple_replace) as [Δ'|] eqn:?; last done.
  rewrite envs_simple_replace_singleton_sound //; simpl.
  by rewrite (into_forall P) (forall_elim x) wand_elim_r.
Qed.

Lemma tac_forall_revert {A} Δ (Φ : A PROP) :
  envs_entails Δ ( a, Φ a) a, envs_entails Δ (Φ a).
Proof. rewrite envs_entails_eq a. by rewrite (forall_elim a). Qed.

Existential

Lemma tac_exist {A} Δ P (Φ : A PROP) :
  FromExist P Φ ( a, envs_entails Δ (Φ a)) envs_entails Δ P.
Proof.
  rewrite envs_entails_eq ⇒ ? [a ?].
  rewrite -(from_exist P). eauto using exist_intro'.
Qed.

Lemma tac_exist_destruct {A} Δ i p j P (Φ : A PROP) Q :
  envs_lookup i Δ = Some (p, P) IntoExist P Φ
  ( a,
    match envs_simple_replace i p (Esnoc Enil j (Φ a)) Δ with
    | Some Δ'envs_entails Δ' Q
    | NoneFalse
    end)
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq ⇒ ?? . rewrite envs_lookup_sound //.
  rewrite (into_exist P) intuitionistically_if_exist sep_exist_r.
  apply exist_elima; specialize ( a) as Hmatch.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:Hrep; last done.
  rewrite envs_simple_replace_singleton_sound' //; simpl. by rewrite wand_elim_r.
Qed.

Modalities

Lemma tac_modal_elim Δ i p p' φ P' P Q Q' :
  envs_lookup i Δ = Some (p, P)
  ElimModal φ p p' P P' Q Q'
  φ
  match envs_replace i p p' (Esnoc Enil i P') Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q'
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_replace _ _ _ _ _) as [Δ'|] eqn:?; last done.
  rewrite envs_entails_eq ⇒ ??? . rewrite envs_replace_singleton_sound //=.
  rewrite . by eapply elim_modal.
Qed.

Accumulate hypotheses

Lemma tac_accu Δ P :
  env_to_prop (env_spatial Δ) = P
  envs_entails Δ P.
Proof.
  rewrite envs_entails_eq=><-.
  rewrite env_to_prop_sound !of_envs_eq and_elim_r sep_elim_r //.
Qed.

Invariants

Lemma tac_inv_elim {X : Type} Δ i j φ p Pinv Pin Pout (Pclose : option (X PROP))
      Q (Q' : X PROP) :
  envs_lookup i Δ = Some (p, Pinv)
  ElimInv φ Pinv Pin Pout Pclose Q Q'
  φ
  ( R,
    match envs_app false (Esnoc Enil j
      (Pin -∗
       ( x, Pout x -∗ pm_option_fun Pclose x -∗? Q' x) -∗
       R
      )%I) (envs_delete false i p Δ)
    with Some Δ''envs_entails Δ'' R | NoneFalse end)
  envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq⇒ ? Hinv ? /(_ Q) Hmatch.
  destruct (envs_app _ _ _) eqn:?; last done.
  rewrite -Hmatch (envs_lookup_sound' _ false) // envs_app_singleton_sound //; simpl.
  apply wand_elim_r', wand_mono; last done. apply wand_intro_r, wand_intro_r.
  rewrite intuitionistically_if_elim -assoc. destruct Pclose; simpl in ×.
  - setoid_rewrite wand_curry. auto.
  - setoid_rewrite <-(right_id emp%I _ (Pout _)). auto.
Qed.

End bi_tactics.

The following private classes are used internally by tac_modal_intro / iModIntro to transform the proofmode environments when introducing a modality.
The class TransformIntuitionisticEnv M C Γin Γout is used to transform the intuitionistic environment using a type class C.
Inputs:
  • Γin : the original environment.
  • M : the modality that the environment should be transformed into.
  • C : PROP PROP Prop : a type class that is used to transform the individual hypotheses. The first parameter is the input and the second parameter is the output.
Outputs:
  • Γout : the resulting environment.
Class TransformIntuitionisticEnv {PROP1 PROP2} (M : modality PROP1 PROP2)
    (C : PROP2 PROP1 Prop) (Γin : env PROP2) (Γout : env PROP1) := {
  transform_intuitionistic_env :
    ( P Q, C P Q P M ( Q))
    ( P Q, M P M Q M (P Q))
     ([∧] Γin) M ( ([∧] Γout));
  transform_intuitionistic_env_wf : env_wf Γin env_wf Γout;
  transform_intuitionistic_env_dom i : Γin !! i = None Γout !! i = None;
}.

Class TransformSpatialEnv {PROP1 PROP2} (M : modality PROP1 PROP2)
    (C : PROP2 PROP1 Prop) (Γin : env PROP2) (Γout : env PROP1)
    (filtered : bool) := {
  transform_spatial_env :
    ( P Q, C P Q P M Q)
    ([∗] Γin) M ([∗] Γout) if filtered then True else emp;
  transform_spatial_env_wf : env_wf Γin env_wf Γout;
  transform_spatial_env_dom i : Γin !! i = None Γout !! i = None;
}.

Inductive IntoModalIntuitionisticEnv {PROP2} : {PROP1} (M : modality PROP1 PROP2)
    (Γin : env PROP2) (Γout : env PROP1), modality_action PROP1 PROP2 Prop :=
  | MIEnvIsEmpty_intuitionistic {PROP1} (M : modality PROP1 PROP2) :
     IntoModalIntuitionisticEnv M Enil Enil MIEnvIsEmpty
  | MIEnvForall_intuitionistic (M : modality PROP2 PROP2) (C : PROP2 Prop) Γ :
     TCForall C (env_to_list Γ)
     IntoModalIntuitionisticEnv M Γ Γ (MIEnvForall C)
  | MIEnvTransform_intuitionistic {PROP1}
       (M : modality PROP1 PROP2) (C : PROP2 PROP1 Prop) Γin Γout :
     TransformIntuitionisticEnv M C Γin Γout
     IntoModalIntuitionisticEnv M Γin Γout (MIEnvTransform C)
  | MIEnvClear_intuitionistic {PROP1 : bi} (M : modality PROP1 PROP2) Γ :
     IntoModalIntuitionisticEnv M Γ Enil MIEnvClear
  | MIEnvId_intuitionistic (M : modality PROP2 PROP2) Γ :
     IntoModalIntuitionisticEnv M Γ Γ MIEnvId.
Existing Class IntoModalIntuitionisticEnv.
Existing Instances MIEnvIsEmpty_intuitionistic MIEnvForall_intuitionistic
  MIEnvTransform_intuitionistic MIEnvClear_intuitionistic MIEnvId_intuitionistic.

Inductive IntoModalSpatialEnv {PROP2} : {PROP1} (M : modality PROP1 PROP2)
    (Γin : env PROP2) (Γout : env PROP1), modality_action PROP1 PROP2 bool Prop :=
  | MIEnvIsEmpty_spatial {PROP1} (M : modality PROP1 PROP2) :
     IntoModalSpatialEnv M Enil Enil MIEnvIsEmpty false
  | MIEnvForall_spatial (M : modality PROP2 PROP2) (C : PROP2 Prop) Γ :
     TCForall C (env_to_list Γ)
     IntoModalSpatialEnv M Γ Γ (MIEnvForall C) false
  | MIEnvTransform_spatial {PROP1}
       (M : modality PROP1 PROP2) (C : PROP2 PROP1 Prop) Γin Γout fi :
     TransformSpatialEnv M C Γin Γout fi
     IntoModalSpatialEnv M Γin Γout (MIEnvTransform C) fi
  | MIEnvClear_spatial {PROP1 : bi} (M : modality PROP1 PROP2) Γ :
     IntoModalSpatialEnv M Γ Enil MIEnvClear false
  | MIEnvId_spatial (M : modality PROP2 PROP2) Γ :
     IntoModalSpatialEnv M Γ Γ MIEnvId false.
Existing Class IntoModalSpatialEnv.
Existing Instances MIEnvIsEmpty_spatial MIEnvForall_spatial
  MIEnvTransform_spatial MIEnvClear_spatial MIEnvId_spatial.

Section tac_modal_intro.
  Context {PROP1 PROP2 : bi} (M : modality PROP1 PROP2).

  Global Instance transform_intuitionistic_env_nil C : TransformIntuitionisticEnv M C Enil Enil.
  Proof.
    split; [|eauto using Enil_wf|done]=> /= ??.
    rewrite !intuitionistically_True_emp -modality_emp //.
  Qed.
  Global Instance transform_intuitionistic_env_snoc (C : PROP2 PROP1 Prop) Γ Γ' i P Q :
    C P Q
    TransformIntuitionisticEnv M C Γ Γ'
    TransformIntuitionisticEnv M C (Esnoc Γ i P) (Esnoc Γ' i Q).
  Proof.
    intros ? [ Hwf Hdom]; split; simpl.
    - intros HC Hand. rewrite intuitionistically_and HC // //.
      by rewrite Hand -intuitionistically_and.
    - inversion 1; constructor; auto.
    - intros j. destruct (ident_beq _ _); naive_solver.
  Qed.
  Global Instance transform_intuitionistic_env_snoc_not (C : PROP2 PROP1 Prop) Γ Γ' i P :
    TransformIntuitionisticEnv M C Γ Γ'
    TransformIntuitionisticEnv M C (Esnoc Γ i P) Γ' | 100.
  Proof.
    intros [ Hwf Hdom]; split; simpl.
    - intros HC Hand. by rewrite and_elim_r .
    - inversion 1; auto.
    - intros j. destruct (ident_beq _ _); naive_solver.
  Qed.

  Global Instance transform_spatial_env_nil C :
    TransformSpatialEnv M C Enil Enil false.
  Proof.
    split; [|eauto using Enil_wf|done]=> /= ?. by rewrite right_id -modality_emp.
  Qed.
  Global Instance transform_spatial_env_snoc (C : PROP2 PROP1 Prop) Γ Γ' i P Q fi :
    C P Q
    TransformSpatialEnv M C Γ Γ' fi
    TransformSpatialEnv M C (Esnoc Γ i P) (Esnoc Γ' i Q) fi.
  Proof.
    intros ? [ Hwf Hdom]; split; simpl.
    - intros HC. by rewrite {1}(HC P) // // assoc modality_sep.
    - inversion 1; constructor; auto.
    - intros j. destruct (ident_beq _ _); naive_solver.
  Qed.

  Global Instance transform_spatial_env_snoc_not
      (C : PROP2 PROP1 Prop) Γ Γ' i P fi fi' :
    TransformSpatialEnv M C Γ Γ' fi
    TCIf (TCEq fi false)
      (TCIf (Affine P) (TCEq fi' false) (TCEq fi' true))
      (TCEq fi' true)
    TransformSpatialEnv M C (Esnoc Γ i P) Γ' fi' | 100.
  Proof.
    intros [ Hwf Hdom] Hif; split; simpl.
    - intros ?. rewrite //. destruct Hif as [-> [? ->| ->]| ->].
      + by rewrite (affine P) left_id.
      + by rewrite right_id comm (True_intro P).
      + by rewrite comm -assoc (True_intro (_ P)%I).
    - inversion 1; auto.
    - intros j. destruct (ident_beq _ _); naive_solver.
  Qed.

The actual introduction tactic
  Lemma tac_modal_intro {A} (sel : A) Γp Γs n Γp' Γs' Q Q' fi :
    FromModal M sel Q' Q
    IntoModalIntuitionisticEnv M Γp Γp' (modality_intuitionistic_action M)
    IntoModalSpatialEnv M Γs Γs' (modality_spatial_action M) fi
    (if fi then Absorbing Q' else TCTrue)
    envs_entails (Envs Γp' Γs' n) Q envs_entails (Envs Γp Γs n) Q'.
  Proof.
    rewrite envs_entails_eq /FromModal !of_envs_eqHQ' HΓp HΓs ? HQ.
    apply pure_elim_l⇒ -[???]. assert (envs_wf (Envs Γp' Γs' n)) as Hwf.
    { split; simpl in ×.
      - destruct HΓp as [| |????? []| |]; eauto using Enil_wf.
      - destruct HΓs as [| |?????? []| |]; eauto using Enil_wf.
      - assert ( i, Γp !! i = None Γp' !! i = None).
        { destruct HΓp as [| |????? []| |]; eauto. }
        assert ( i, Γs !! i = None Γs' !! i = None).
        { destruct HΓs as [| |?????? []| |]; eauto. }
        naive_solver. }
    assert ( [∧] Γp M ( [∧] Γp'))%I as HMp.
    { remember (modality_intuitionistic_action M).
      destruct HΓp as [?|M C Γp ?%TCForall_Forall|? M C Γp Γp' []|? M Γp|M Γp]; simpl.
      - rewrite {1}intuitionistically_elim_emp (modality_emp M)
          intuitionistically_True_emp //.
      - eauto using modality_intuitionistic_forall_big_and.
      - eauto using modality_intuitionistic_transform,
          modality_and_transform.
      - by rewrite {1}intuitionistically_elim_emp (modality_emp M)
          intuitionistically_True_emp.
      - eauto using modality_intuitionistic_id. }
    move: HQ'; rewrite -HQ pure_True // left_id HMpHQ' {HQ Hwf HMp}.
    remember (modality_spatial_action M).
    destruct HΓs as [?|M C Γs ?%TCForall_Forall|? M C Γs Γs' fi []|? M Γs|M Γs]; simpl.
    - by rewrite -HQ' /= !right_id.
    - rewrite -HQ' {1}(modality_spatial_forall_big_sep _ _ Γs) //.
      by rewrite modality_sep.
    - destruct fi.
      + rewrite -(absorbing Q') /bi_absorbingly -HQ' (comm _ True%I).
        rewrite -modality_sep -assoc. apply sep_mono_r.
        eauto using modality_spatial_transform.
      + rewrite -HQ' -modality_sep. apply sep_mono_r.
        rewrite -(right_id emp%I bi_sep (M _)).
        eauto using modality_spatial_transform.
    - rewrite -HQ' /= right_id comm -{2}(modality_spatial_clear M) //.
      by rewrite (True_intro ([∗] Γs)%I).
    - rewrite -HQ' {1}(modality_spatial_id M ([∗] Γs)%I) //.
      by rewrite -modality_sep.
  Qed.
End tac_modal_intro.

Section sbi_tactics.
Context {PROP : sbi}.
Implicit Types Γ : env PROP.
Implicit Types Δ : envs PROP.
Implicit Types P Q : PROP.

Rewriting

Lemma tac_rewrite Δ i p Pxy d Q :
  envs_lookup i Δ = Some (p, Pxy)
   {A : ofeT} (x y : A) (Φ : A PROP),
    IntoInternalEq Pxy x y
    (Q ⊣⊢ Φ (if d is Left then y else x))
    NonExpansive Φ
    envs_entails Δ (Φ (if d is Left then x else y)) envs_entails Δ Q.
Proof.
  intros ? A x y ? HPxy → ?. rewrite envs_entails_eq.
  apply internal_eq_rewrite'; auto. rewrite {1}envs_lookup_sound //.
  rewrite (into_internal_eq Pxy x y) intuitionistically_if_elim sep_elim_l.
  destruct d; auto using internal_eq_sym.
Qed.

Lemma tac_rewrite_in Δ i p Pxy j q P d Q :
  envs_lookup i Δ = Some (p, Pxy)
  envs_lookup j Δ = Some (q, P)
   {A : ofeT} (x y : A) (Φ : A PROP),
    IntoInternalEq Pxy x y
    (P ⊣⊢ Φ (if d is Left then y else x))
    NonExpansive Φ
    match envs_simple_replace j q (Esnoc Enil j (Φ (if d is Left then x else y))) Δ with
    | NoneFalse
    | Some Δ'envs_entails Δ' Q
    end
    envs_entails Δ Q.
Proof.
  rewrite envs_entails_eq /IntoInternalEq ⇒ ?? A x y Φ HPxy HP ? Hentails.
  destruct (envs_simple_replace _ _ _ _) as [Δ'|] eqn:?; last done. rewrite -Hentails.
  rewrite -(idemp bi_and (of_envs Δ)) {2}(envs_lookup_sound _ i) //.
  rewrite (envs_simple_replace_singleton_sound _ _ j) //=.
  rewrite HP HPxy (intuitionistically_if_elim _ (_ _)%I) sep_elim_l.
  rewrite persistent_and_affinely_sep_r -assoc. apply wand_elim_r'.
  rewrite -persistent_and_affinely_sep_r. apply impl_elim_r'. destruct d.
  - apply (internal_eq_rewrite x y (λ y, □?q Φ y -∗ of_envs Δ')%I). solve_proper.
  - rewrite internal_eq_sym.
    eapply (internal_eq_rewrite y x (λ y, □?q Φ y -∗ of_envs Δ')%I). solve_proper.
Qed.

Later

The class MaybeIntoLaterNEnvs is used by tactics that need to introduce laters, e.g. the symbolic execution tactics.
Class MaybeIntoLaterNEnvs (n : nat) (Δ1 Δ2 : envs PROP) := {
  into_later_intuitionistic :
    TransformIntuitionisticEnv (modality_laterN n) (MaybeIntoLaterN false n)
      (env_intuitionistic Δ1) (env_intuitionistic Δ2);
  into_later_spatial :
    TransformSpatialEnv (modality_laterN n)
      (MaybeIntoLaterN false n) (env_spatial Δ1) (env_spatial Δ2) false
}.

Global Instance into_laterN_envs n Γp1 Γp2 Γs1 Γs2 m :
  TransformIntuitionisticEnv (modality_laterN n) (MaybeIntoLaterN false n) Γp1 Γp2
  TransformSpatialEnv (modality_laterN n) (MaybeIntoLaterN false n) Γs1 Γs2 false
  MaybeIntoLaterNEnvs n (Envs Γp1 Γs1 m) (Envs Γp2 Γs2 m).
Proof. by split. Qed.

Lemma into_laterN_env_sound n Δ1 Δ2 :
  MaybeIntoLaterNEnvs n Δ1 Δ2 of_envs Δ1 ▷^n (of_envs Δ2).
Proof.
  intros [[Hp ??] [Hs ??]]; rewrite !of_envs_eq /= !laterN_and !laterN_sep.
  rewrite -{1}laterN_intro. apply and_mono, sep_mono.
  - apply pure_mono; destruct 1; constructor; naive_solver.
  - apply Hp; rewrite /= /MaybeIntoLaterN.
    + intros P Q →. by rewrite laterN_intuitionistically_2.
    + intros P Q. by rewrite laterN_and.
  - by rewrite Hs //= right_id.
Qed.

Lemma tac_löb Δ i Q :
  env_spatial_is_nil Δ = true
  match envs_app true (Esnoc Enil i ( Q)%I) Δ with
  | NoneFalse
  | Some Δ'envs_entails Δ' Q
  end
  envs_entails Δ Q.
Proof.
  destruct (envs_app _ _ _) eqn:?; last done.
  rewrite envs_entails_eq ⇒ ? HQ.
  rewrite (env_spatial_is_nil_intuitionistically Δ) //.
  rewrite -(persistently_and_emp_elim Q). apply and_intro; first apply: affine.
  rewrite -(löb (<pers> Q)%I) later_persistently. apply impl_intro_l.
  rewrite envs_app_singleton_sound //; simpl; rewrite HQ.
  rewrite persistently_and_intuitionistically_sep_l -{1}intuitionistically_idemp.
  rewrite intuitionistically_sep_2 wand_elim_r intuitionistically_into_persistently_1 //.
Qed.
End sbi_tactics.