coq_tactics.v 41.2 KB
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From iris.base_logic Require Export base_logic.
From iris.base_logic Require Import big_op tactics.
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From iris.proofmode Require Export base environments classes.
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Set Default Proof Using "Type".
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Import uPred.
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Import env_notations.
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Local Notation "b1 && b2" := (if b1 then b2 else false) : bool_scope.

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Record envs (M : ucmraT) :=
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  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
}.

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Definition of_envs {M} (Δ : envs M) : uPred M :=
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  (envs_wf Δ⌝   [] env_persistent Δ  [] env_spatial Δ)%I.
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Instance: Params (@of_envs) 1.

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Definition envs_entails {M} (Δ : envs M) (Q : uPred M) : Prop :=
  of_envs Δ  Q.
Arguments envs_entails {_} _ _%I.
Typeclasses Opaque envs_entails.
Instance: Params (@envs_entails) 1.

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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)
}.
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Definition envs_dom {M} (Δ : envs M) : list ident :=
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  env_dom (env_persistent Δ) ++ env_dom (env_spatial Δ).
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Definition envs_lookup {M} (i : ident) (Δ : envs M) : option (bool * uPred M) :=
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  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.

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Definition envs_delete {M} (i : ident) (p : bool) (Δ : envs M) : envs M :=
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  let (Γp,Γs) := Δ in
  match p with
  | true => Envs (env_delete i Γp) Γs | false => Envs Γp (env_delete i Γs)
  end.

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Definition envs_lookup_delete {M} (i : ident)
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    (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.

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Fixpoint envs_lookup_delete_list {M} (js : list ident) (remove_persistent : bool)
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    (Δ : envs M) : option (bool * list (uPred M) * envs M) :=
  match js with
  | [] => Some (true, [], Δ)
  | j :: js =>
     '(p,P,Δ')  envs_lookup_delete j Δ;
     let Δ' := if p then (if remove_persistent then Δ' else Δ) else Δ' in
     '(q,Hs,Δ'')  envs_lookup_delete_list js remove_persistent Δ';
     Some (p && q, P :: Hs, Δ'')
  end.

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Definition envs_snoc {M} (Δ : envs M)
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    (p : bool) (j : ident) (P : uPred M) : envs M :=
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  let (Γp,Γs) := Δ in
  if p then Envs (Esnoc Γp j P) Γs else Envs Γp (Esnoc Γs j P).

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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.

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Definition envs_simple_replace {M} (i : ident) (p : bool) (Γ : env (uPred M))
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    (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.

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Definition envs_replace {M} (i : ident) (p q : bool) (Γ : env (uPred M))
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    (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).

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Definition env_spatial_is_nil {M} (Δ : envs M) :=
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  if env_spatial Δ is Enil then true else false.

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Definition envs_clear_spatial {M} (Δ : envs M) : envs M :=
  Envs (env_persistent Δ) Enil.

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Definition envs_clear_persistent {M} (Δ : envs M) : envs M :=
  Envs Enil (env_spatial Δ).

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Fixpoint envs_split_go {M}
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    (js : list ident) (Δ1 Δ2 : envs M) : option (envs M * envs M) :=
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  match js with
  | [] => Some (Δ1, Δ2)
  | j :: js =>
     '(p,P,Δ1')  envs_lookup_delete j Δ1;
     if p then envs_split_go js Δ1 Δ2 else
     envs_split_go js Δ1' (envs_snoc Δ2 false j P)
  end.
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(* if [d = Right] then [result = (remaining hyps, hyps named js)] and
   if [d = Left] then [result = (hyps named js, remaining hyps)] *)
Definition envs_split {M} (d : direction)
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    (js : list ident) (Δ : envs M) : option (envs M * envs M) :=
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  '(Δ1,Δ2)  envs_split_go js Δ (envs_clear_spatial Δ);
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  if d is Right then Some (Δ1,Δ2) else Some (Δ2,Δ1).
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(* Coq versions of the tactics *)
Section tactics.
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Context {M : ucmraT}.
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Implicit Types Γ : env (uPred M).
Implicit Types Δ : envs M.
Implicit Types P Q : uPred M.

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Lemma of_envs_def Δ :
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  of_envs Δ = (envs_wf Δ⌝   [] env_persistent Δ  [] env_spatial Δ)%I.
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Proof. done. Qed.

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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 :
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  envs_lookup i Δ = Some (p,P) 
  of_envs Δ  ?p P  of_envs (envs_delete i p Δ).
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Proof.
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  rewrite /envs_lookup /envs_delete /of_envs=>?; apply pure_elim_sep_l=> Hwf.
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  destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
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  - rewrite (env_lookup_perm Γp) //= persistently_sep.
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    ecancel [ [] _;  P; [] Γs]%I; apply pure_intro.
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    destruct Hwf; constructor;
      naive_solver eauto using env_delete_wf, env_delete_fresh.
  - destruct (Γs !! i) eqn:?; simplify_eq/=.
    rewrite (env_lookup_perm Γs) //=.
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    ecancel [ [] _; P; [] (env_delete _ _)]%I; apply pure_intro.
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    destruct Hwf; constructor;
      naive_solver eauto using env_delete_wf, env_delete_fresh.
Qed.
Lemma envs_lookup_sound' Δ i p P :
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  envs_lookup i Δ = Some (p,P) 
  of_envs Δ  P  of_envs (envs_delete i p Δ).
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Proof. intros. rewrite envs_lookup_sound //. by rewrite persistently_if_elim. Qed.
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Lemma envs_lookup_persistent_sound Δ i P :
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  envs_lookup i Δ = Some (true,P)  of_envs Δ   P  of_envs Δ.
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Proof.
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  intros. apply (persistently_entails_l _ _). by rewrite envs_lookup_sound // sep_elim_l.
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Qed.

Lemma envs_lookup_split Δ i p P :
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  envs_lookup i Δ = Some (p,P)  of_envs Δ  ?p P  (?p P - of_envs Δ).
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Proof.
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  rewrite /envs_lookup /of_envs=>?; apply pure_elim_sep_l=> Hwf.
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  destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
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  - rewrite (env_lookup_perm Γp) //= persistently_sep.
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    rewrite pure_True // left_id.
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    cancel [ P]%I. apply wand_intro_l. solve_sep_entails.
  - destruct (Γs !! i) eqn:?; simplify_eq/=.
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    rewrite (env_lookup_perm Γs) //=. rewrite pure_True // left_id.
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    cancel [P]. apply wand_intro_l. solve_sep_entails.
Qed.

Lemma envs_lookup_delete_sound Δ Δ' i p P :
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  envs_lookup_delete i Δ = Some (p,P,Δ')  of_envs Δ  ?p P  of_envs Δ'.
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Proof. intros [? ->]%envs_lookup_delete_Some. by apply envs_lookup_sound. Qed.
Lemma envs_lookup_delete_sound' Δ Δ' i p P :
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  envs_lookup_delete i Δ = Some (p,P,Δ')  of_envs Δ  P  of_envs Δ'.
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Proof. intros [? ->]%envs_lookup_delete_Some. by apply envs_lookup_sound'. Qed.

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Lemma envs_lookup_delete_list_sound Δ Δ' js rp p Ps :
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  envs_lookup_delete_list js rp Δ = Some (p, Ps,Δ') 
  of_envs Δ  ?p [] Ps  of_envs Δ'.
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Proof.
  revert Δ Δ' p Ps. induction js as [|j js IH]=> Δ Δ'' p Ps ?; simplify_eq/=.
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  { by rewrite persistently_pure left_id. }
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  destruct (envs_lookup_delete j Δ) as [[[q1 P] Δ']|] eqn:Hj; simplify_eq/=.
  apply envs_lookup_delete_Some in Hj as [Hj ->].
  destruct (envs_lookup_delete_list js rp _) as [[[q2 Ps'] ?]|] eqn:?; simplify_eq/=.
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  rewrite persistently_if_sep -assoc. destruct q1; simpl.
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  - destruct rp.
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    + rewrite envs_lookup_sound //; simpl. by rewrite IH // (persistently_elim_if q2).
    + rewrite envs_lookup_persistent_sound //. by rewrite IH // (persistently_elim_if q2).
  - rewrite envs_lookup_sound // IH //; simpl. by rewrite persistently_if_elim.
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Qed.

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Lemma envs_lookup_snoc Δ i p P :
  envs_lookup i Δ = None  envs_lookup i (envs_snoc Δ p i P) = Some (p, P).
Proof.
  rewrite /envs_lookup /envs_snoc=> ?.
  destruct Δ as [Γp Γs], p, (Γp !! i); simplify_eq; by rewrite env_lookup_snoc.
Qed.
Lemma envs_lookup_snoc_ne Δ i j p P :
  i  j  envs_lookup i (envs_snoc Δ p j P) = envs_lookup i Δ.
Proof.
  rewrite /envs_lookup /envs_snoc=> ?.
  destruct Δ as [Γp Γs], p; simplify_eq; by rewrite env_lookup_snoc_ne.
Qed.

Lemma envs_snoc_sound Δ p i P :
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  envs_lookup i Δ = None  of_envs Δ  ?p P - of_envs (envs_snoc Δ p i P).
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Proof.
  rewrite /envs_lookup /envs_snoc /of_envs=> ?; apply pure_elim_sep_l=> Hwf.
  destruct Δ as [Γp Γs], (Γp !! i) eqn:?, (Γs !! i) eqn:?; simplify_eq/=.
  apply wand_intro_l; destruct p; simpl.
  - apply sep_intro_True_l; [apply pure_intro|].
    + destruct Hwf; constructor; simpl; eauto using Esnoc_wf.
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      intros j; destruct (ident_beq_reflect j i); naive_solver.
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    + by rewrite persistently_sep assoc.
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  - apply sep_intro_True_l; [apply pure_intro|].
    + destruct Hwf; constructor; simpl; eauto using Esnoc_wf.
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      intros j; destruct (ident_beq_reflect j i); naive_solver.
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    + solve_sep_entails.
Qed.

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Lemma envs_app_sound Δ Δ' p Γ :
  envs_app p Γ Δ = Some Δ'  of_envs Δ  ?p [] Γ - of_envs Δ'.
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Proof.
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  rewrite /of_envs /envs_app=> ?; apply pure_elim_sep_l=> Hwf.
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  destruct Δ as [Γp Γs], p; simplify_eq/=.
  - destruct (env_app Γ Γs) eqn:Happ,
      (env_app Γ Γp) as [Γp'|] eqn:?; simplify_eq/=.
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    apply wand_intro_l, sep_intro_True_l; [apply pure_intro|].
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    + 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') //.
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      rewrite big_opL_app persistently_sep. solve_sep_entails.
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  - destruct (env_app Γ Γp) eqn:Happ,
      (env_app Γ Γs) as [Γs'|] eqn:?; simplify_eq/=.
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    apply wand_intro_l, sep_intro_True_l; [apply pure_intro|].
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    + 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.
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    + rewrite (env_app_perm _ _ Γs') // big_opL_app. solve_sep_entails.
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Qed.

Lemma envs_simple_replace_sound' Δ Δ' i p Γ :
  envs_simple_replace i p Γ Δ = Some Δ' 
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  of_envs (envs_delete i p Δ)  ?p [] Γ - of_envs Δ'.
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Proof.
  rewrite /envs_simple_replace /envs_delete /of_envs=> ?.
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  apply pure_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], p; simplify_eq/=.
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  - destruct (env_app Γ Γs) eqn:Happ,
      (env_replace i Γ Γp) as [Γp'|] eqn:?; simplify_eq/=.
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    apply wand_intro_l, sep_intro_True_l; [apply pure_intro|].
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    + 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') //.
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      rewrite big_opL_app persistently_sep. solve_sep_entails.
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  - destruct (env_app Γ Γp) eqn:Happ,
      (env_replace i Γ Γs) as [Γs'|] eqn:?; simplify_eq/=.
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    apply wand_intro_l, sep_intro_True_l; [apply pure_intro|].
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    + 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.
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    + rewrite (env_replace_perm _ _ Γs') // big_opL_app. solve_sep_entails.
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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' 
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  of_envs Δ  ?p P  (?p [] Γ - of_envs Δ').
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Proof. intros. by rewrite envs_lookup_sound// envs_simple_replace_sound'//. Qed.

Lemma envs_replace_sound' Δ Δ' i p q Γ :
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  envs_replace i p q Γ Δ = Some Δ' 
  of_envs (envs_delete i p Δ)  ?q [] Γ - of_envs Δ'.
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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' 
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  of_envs Δ  ?p P  (?q [] Γ - of_envs Δ').
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Proof. intros. by rewrite envs_lookup_sound// envs_replace_sound'//. Qed.

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Lemma envs_lookup_envs_clear_spatial Δ j :
  envs_lookup j (envs_clear_spatial Δ)
  = '(p,P)  envs_lookup j Δ; if p then Some (p,P) else None.
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Proof.
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  rewrite /envs_lookup /envs_clear_spatial.
  destruct Δ as [Γp Γs]; simpl; destruct (Γp !! j) eqn:?; simplify_eq/=; auto.
  by destruct (Γs !! j).
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Qed.

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Lemma envs_clear_spatial_sound Δ :
  of_envs Δ  of_envs (envs_clear_spatial Δ)  [] env_spatial Δ.
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Proof.
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  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].
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  destruct Hwf; constructor; simpl; auto using Enil_wf.
Qed.

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Lemma env_spatial_is_nil_persistent Δ :
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  env_spatial_is_nil Δ = true  Persistent (of_envs Δ).
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Proof. intros; destruct Δ as [? []]; simplify_eq/=; apply _. Qed.
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Hint Immediate env_spatial_is_nil_persistent : typeclass_instances.
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Lemma envs_lookup_envs_delete Δ i p P :
  envs_wf Δ 
  envs_lookup i Δ = Some (p,P)  envs_lookup i (envs_delete i p Δ) = None.
Proof.
  rewrite /envs_lookup /envs_delete=> -[?? Hdisj] Hlookup.
  destruct Δ as [Γp Γs], p; simplify_eq/=.
  - rewrite env_lookup_env_delete //. revert Hlookup.
    destruct (Hdisj i) as [->| ->]; [|done]. by destruct (Γs !! _).
  - rewrite env_lookup_env_delete //. by destruct (Γp !! _).
Qed.
Lemma envs_lookup_envs_delete_ne Δ i j p :
  i  j  envs_lookup i (envs_delete j p Δ) = envs_lookup i Δ.
Proof.
  rewrite /envs_lookup /envs_delete=> ?. destruct Δ as [Γp Γs],p; simplify_eq/=.
  - by rewrite env_lookup_env_delete_ne.
  - destruct (Γp !! i); simplify_eq/=; by rewrite ?env_lookup_env_delete_ne.
Qed.

Lemma envs_split_go_sound js Δ1 Δ2 Δ1' Δ2' :
  ( j P, envs_lookup j Δ1 = Some (false, P)  envs_lookup j Δ2 = None) 
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  envs_split_go js Δ1 Δ2 = Some (Δ1',Δ2') 
  of_envs Δ1  of_envs Δ2  of_envs Δ1'  of_envs Δ2'.
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Proof.
  revert Δ1 Δ2 Δ1' Δ2'.
  induction js as [|j js IH]=> Δ1 Δ2 Δ1' Δ2' Hlookup HΔ; simplify_eq/=; [done|].
  apply pure_elim with (envs_wf Δ1); [unfold of_envs; solve_sep_entails|]=> Hwf.
  destruct (envs_lookup_delete j Δ1)
    as [[[[] P] Δ1'']|] eqn:Hdel; simplify_eq; auto.
  apply envs_lookup_delete_Some in Hdel as [??]; subst.
  rewrite envs_lookup_sound //; rewrite /= (comm _ P) -assoc.
  rewrite -(IH _ _ _ _ _ HΔ); last first.
  { intros j' P'; destruct (decide (j = j')) as [->|].
    - by rewrite (envs_lookup_envs_delete _ _ _ P).
    - rewrite envs_lookup_envs_delete_ne // envs_lookup_snoc_ne //. eauto. }
  rewrite (envs_snoc_sound Δ2 false j P) /= ?wand_elim_r; eauto.
Qed.
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Lemma envs_split_sound Δ d js Δ1 Δ2 :
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  envs_split d js Δ = Some (Δ1,Δ2)  of_envs Δ  of_envs Δ1  of_envs Δ2.
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Proof.
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  rewrite /envs_split=> ?. rewrite -(idemp uPred_and (of_envs Δ)).
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  rewrite {2}envs_clear_spatial_sound sep_elim_l and_sep_r.
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  destruct (envs_split_go _ _) as [[Δ1' Δ2']|] eqn:HΔ; [|done].
  apply envs_split_go_sound in HΔ as ->; last first.
  { intros j P. by rewrite envs_lookup_envs_clear_spatial=> ->. }
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  destruct d; simplify_eq; solve_sep_entails.
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Qed.

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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 *.
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  apply pure_elim_sep_l=>Hwf. apply sep_intro_True_l.
  - destruct Hwf; apply pure_intro; constructor;
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      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.
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  intros Δ1 Δ2 HΔ; apply (anti_symm ()); apply of_envs_mono;
    eapply (envs_Forall2_impl ()); [| |symmetry|]; eauto using equiv_entails.
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Qed.
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Global Instance Envs_proper (R : relation (uPred M)) :
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  Proper (env_Forall2 R ==> env_Forall2 R ==> envs_Forall2 R) (@Envs M).
Proof. by constructor. Qed.

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Global Instance envs_entails_proper :
  Proper (envs_Forall2 () ==> () ==> iff) (@envs_entails M).
Proof. solve_proper. Qed.
Global Instance envs_entails_flip_mono :
  Proper (envs_Forall2 () ==> flip () ==> flip impl) (@envs_entails M).
Proof. rewrite /envs_entails=> Δ1 Δ2 ? P1 P2 <- <-. by f_equiv. Qed.

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(** * Adequacy *)
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Lemma tac_adequate P : envs_entails (Envs Enil Enil) P  P.
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Proof.
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  rewrite /envs_entails=> <-. rewrite /of_envs /= persistently_pure !right_id.
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  apply pure_intro; repeat constructor.
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Qed.

(** * Basic rules *)
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Lemma tac_assumption Δ i p P Q :
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  envs_lookup i Δ = Some (p,P)  FromAssumption p P Q 
  envs_entails Δ Q.
Proof. intros. by rewrite /envs_entails envs_lookup_sound // sep_elim_l. Qed.
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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' 
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  envs_entails Δ' Q 
  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails=> ?? <-. rewrite envs_simple_replace_sound //.
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  destruct p; simpl; by rewrite right_id wand_elim_r.
Qed.
Lemma tac_clear Δ Δ' i p P Q :
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  envs_lookup_delete i Δ = Some (p,P,Δ') 
  envs_entails Δ' Q  envs_entails Δ Q.
Proof.
  rewrite /envs_entails=> ? <-. by rewrite envs_lookup_delete_sound // sep_elim_r.
Qed.
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(** * False *)
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Lemma tac_ex_falso Δ Q : envs_entails Δ False  envs_entails Δ Q.
Proof. by rewrite /envs_entails -(False_elim Q). Qed.
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(** * Pure *)
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Lemma tac_pure_intro Δ Q φ : FromPure Q φ  φ  envs_entails Δ Q.
Proof. intros ??. rewrite /envs_entails -(from_pure Q). by apply pure_intro. Qed.
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Lemma tac_pure Δ Δ' i p P φ Q :
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  envs_lookup_delete i Δ = Some (p, P, Δ')  IntoPure P φ 
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  (φ  envs_entails Δ' Q)  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails=> ?? HQ. rewrite envs_lookup_delete_sound' //; simpl.
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  rewrite (into_pure P); by apply pure_elim_sep_l.
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Qed.

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Lemma tac_pure_revert Δ φ Q : envs_entails Δ (⌜φ⌝  Q)  (φ  envs_entails Δ Q).
Proof. rewrite /envs_entails. intros HΔ ?. by rewrite HΔ pure_True // left_id. Qed.
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(** * Later *)
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Class IntoLaterNEnv (n : nat) (Γ1 Γ2 : env (uPred M)) :=
  into_laterN_env : env_Forall2 (IntoLaterN n) Γ1 Γ2.
Class IntoLaterNEnvs (n : nat) (Δ1 Δ2 : envs M) := {
  into_later_persistent: IntoLaterNEnv n (env_persistent Δ1) (env_persistent Δ2);
  into_later_spatial: IntoLaterNEnv n (env_spatial Δ1) (env_spatial Δ2)
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}.

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Global Instance into_laterN_env_nil n : IntoLaterNEnv n Enil Enil.
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Proof. constructor. Qed.
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Global Instance into_laterN_env_snoc n Γ1 Γ2 i P Q :
  IntoLaterNEnv n Γ1 Γ2  IntoLaterN n P Q 
  IntoLaterNEnv n (Esnoc Γ1 i P) (Esnoc Γ2 i Q).
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Proof. by constructor. Qed.

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Global Instance into_laterN_envs n Γp1 Γp2 Γs1 Γs2 :
  IntoLaterNEnv n Γp1 Γp2  IntoLaterNEnv n Γs1 Γs2 
  IntoLaterNEnvs n (Envs Γp1 Γs1) (Envs Γp2 Γs2).
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Proof. by split. Qed.
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Lemma into_laterN_env_sound n Δ1 Δ2 :
  IntoLaterNEnvs n Δ1 Δ2  of_envs Δ1  ^n (of_envs Δ2).
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Proof.
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  intros [Hp Hs]; rewrite /of_envs /= !laterN_sep -persistently_laterN.
  repeat apply sep_mono; try apply persistently_mono.
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  - rewrite -laterN_intro; apply pure_mono; destruct 1; constructor;
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      naive_solver eauto using env_Forall2_wf, env_Forall2_fresh.
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  - induction Hp; rewrite /= ?laterN_sep. apply laterN_intro. by apply sep_mono.
  - induction Hs; rewrite /= ?laterN_sep. apply laterN_intro. by apply sep_mono.
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Qed.

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Lemma tac_next Δ Δ' n Q Q' :
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  FromLaterN n Q Q'  IntoLaterNEnvs n Δ Δ' 
  envs_entails Δ' Q'  envs_entails Δ Q.
Proof.
  rewrite /envs_entails=> ?? HQ.
  by rewrite -(from_laterN n Q) into_laterN_env_sound HQ.
Qed.
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Lemma tac_löb Δ Δ' i Q :
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  env_spatial_is_nil Δ = true 
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  envs_app true (Esnoc Enil i ( Q)%I) Δ = Some Δ' 
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  envs_entails Δ' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails=> ?? HQ.
  rewrite -(persistently_elim Q) -(löb ( Q)) -persistently_later.
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  apply impl_intro_l, (persistently_intro _ _).
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  rewrite envs_app_sound //; simpl.
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  by rewrite right_id persistently_and_sep_l wand_elim_r HQ.
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Qed.

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(** * Persistence and plainness modality *)
Class IntoPlainEnv (Γ1 Γ2 : env (uPred M)) := {
  into_plain_env_subenv : env_subenv Γ2 Γ1;
  into_plain_env_plain : Plain ([] Γ2);
}.
Class IntoPersistentEnvs (p : bool) (Δ1 Δ2 : envs M) := {
  into_persistent_envs_persistent :
    if p then IntoPlainEnv (env_persistent Δ1) (env_persistent Δ2)
    else env_persistent Δ1 = env_persistent Δ2;
  into_persistent_envs_spatial : env_spatial Δ2 = Enil
}.

Global Instance into_plain_env_nil : IntoPlainEnv Enil Enil.
Proof. constructor. constructor. simpl; apply _. Qed.
Global Instance into_plain_env_snoc_plain Γ1 Γ2 i P :
  Plain P  IntoPlainEnv Γ1 Γ2 
  IntoPlainEnv (Esnoc Γ1 i P) (Esnoc Γ2 i P) | 1.
Proof. intros ? [??]; constructor. by constructor. simpl; apply _. Qed.
Global Instance into_plain_env_snoc_skip Γ1 Γ2 i P :
  IntoPlainEnv Γ1 Γ2  IntoPlainEnv (Esnoc Γ1 i P) Γ2 | 2.
Proof. intros [??]; constructor. by constructor. done. Qed.

Global Instance into_persistent_envs_false Γp Γs :
  IntoPersistentEnvs false (Envs Γp Γs) (Envs Γp Enil).
Proof. by split. Qed.
Global Instance into_persistent_envs_true Γp1 Γp2 Γs1 :
  IntoPlainEnv Γp1 Γp2 
  IntoPersistentEnvs true (Envs Γp1 Γs1) (Envs Γp2 Enil).
Proof. by split. Qed.

Lemma into_persistent_envs_sound (p : bool) Δ1 Δ2 :
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  IntoPersistentEnvs p Δ1 Δ2 
  of_envs Δ1  (if p then  of_envs Δ2 else  of_envs Δ2).
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Proof.
  rewrite /of_envs. destruct Δ1 as [Γp1 Γs1], Δ2 as [Γp2 Γs2]=> -[/= Hp ->].
  apply pure_elim_sep_l=> Hwf. rewrite sep_elim_l. destruct p; simplify_eq/=.
  - destruct Hp. rewrite right_id plainly_sep plainly_pure.
    apply sep_intro_True_l; [apply pure_intro|].
    + destruct Hwf; constructor; eauto using Enil_wf, env_subenv_wf.
    + rewrite persistently_elim plainly_persistently plainly_plainly.
      by apply big_sepL_submseteq, sublist_submseteq, env_to_list_subenv_proper.
  - rewrite right_id persistently_sep persistently_pure.
    apply sep_intro_True_l; [apply pure_intro|by rewrite persistent_persistently].
    destruct Hwf; constructor; simpl; eauto using Enil_wf.
Qed.

Lemma tac_always_intro Δ Δ' p Q Q' :
  FromAlways p Q' Q 
  IntoPersistentEnvs p Δ Δ' 
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  envs_entails Δ' Q  envs_entails Δ Q'.
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Proof.
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  rewrite /envs_entails=> ?? HQ.
  rewrite into_persistent_envs_sound -(from_always _ Q').
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  destruct p; auto using persistently_mono, plainly_mono.
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Qed.
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Lemma tac_persistent Δ Δ' i p P P' Q :
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  envs_lookup i Δ = Some (p, P) 
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  IntoPersistent p P P' 
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  envs_replace i p true (Esnoc Enil i P') Δ = Some Δ' 
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  envs_entails Δ' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails=> ? HP ? <-. rewrite envs_replace_sound //; simpl.
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  by rewrite right_id (into_persistent _ P) wand_elim_r.
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Qed.

(** * Implication and wand *)
Lemma tac_impl_intro Δ Δ' i P Q :
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  (if env_spatial_is_nil Δ then TCTrue else Persistent P) 
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  envs_app false (Esnoc Enil i P) Δ = Some Δ' 
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  envs_entails Δ' Q  envs_entails Δ (P  Q).
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Proof.
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  rewrite /envs_entails=> ?? <-. destruct (env_spatial_is_nil Δ) eqn:?.
  - rewrite (persistent (of_envs Δ)) envs_app_sound //; simpl.
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    by rewrite right_id -persistently_impl_wand persistently_elim.
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  - apply impl_intro_l. rewrite envs_app_sound //; simpl.
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    by rewrite and_sep_l right_id wand_elim_r.
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Qed.
Lemma tac_impl_intro_persistent Δ Δ' i P P' Q :
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  IntoPersistent false P P' 
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  envs_app true (Esnoc Enil i P') Δ = Some Δ' 
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  envs_entails Δ' Q  envs_entails Δ (P  Q).
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Proof.
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  rewrite /envs_entails=> ?? HQ.
  rewrite envs_app_sound //=; simpl. apply impl_intro_l.
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  rewrite (_ : P = ?false P) // (into_persistent false P).
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  by rewrite right_id persistently_and_sep_l wand_elim_r.
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Qed.
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Lemma tac_impl_intro_drop Δ P Q : envs_entails Δ Q  envs_entails Δ (P  Q).
Proof. rewrite /envs_entails=> ?. apply impl_intro_l. by rewrite and_elim_r. Qed.
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Lemma tac_wand_intro Δ Δ' i P Q :
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  envs_app false (Esnoc Enil i P) Δ = Some Δ' 
  envs_entails Δ' Q  envs_entails Δ (P - Q).
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Proof.
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  rewrite /envs_entails=> ? HQ.
  rewrite envs_app_sound //; simpl. by rewrite right_id HQ.
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Qed.
Lemma tac_wand_intro_persistent Δ Δ' i P P' Q :
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  IntoPersistent false P P' 
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  envs_app true (Esnoc Enil i P') Δ = Some Δ' 
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  envs_entails Δ' Q  envs_entails Δ (P - Q).
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Proof.
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  rewrite /envs_entails => ?? <-. rewrite envs_app_sound //; simpl.
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  rewrite right_id. by apply wand_mono.
Qed.
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Lemma tac_wand_intro_drop Δ P Q : envs_entails Δ Q  envs_entails Δ (P - Q).
Proof. rewrite /envs_entails=> <-. apply wand_intro_l. by rewrite sep_elim_r. Qed.
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(* 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') 
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  envs_lookup j (if p then Δ else Δ') = Some (q, R) 
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  IntoWand p R P1 P2 
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  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'' 
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  envs_entails Δ'' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails. intros [? ->]%envs_lookup_delete_Some ??? <-. destruct p.
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  - rewrite envs_lookup_persistent_sound // envs_simple_replace_sound //; simpl.
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    rewrite right_id assoc (into_wand _ R) /=. destruct q; simpl.
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    + by rewrite persistently_wand persistent_persistently !wand_elim_r.
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    + by rewrite !wand_elim_r.
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  - rewrite envs_lookup_sound //; simpl.
    rewrite envs_lookup_sound // (envs_replace_sound' _ Δ'') //; simpl.
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    by rewrite right_id assoc (into_wand _ R) persistently_if_elim wand_elim_r wand_elim_r.
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Qed.

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Lemma tac_specialize_assert Δ Δ' Δ1 Δ2' j q neg js R P1 P2 P1' Q :
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  envs_lookup_delete j Δ = Some (q, R, Δ') 
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  IntoWand false R P1 P2  ElimModal P1' P1 Q Q 
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  ('(Δ1,Δ2)  envs_split (if neg is true then Right else Left) js Δ';
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    Δ2'  envs_app false (Esnoc Enil j P2) Δ2;
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    Some (Δ1,Δ2')) = Some (Δ1,Δ2')  (* does not preserve position of [j] *)
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  envs_entails Δ1 P1'  envs_entails Δ2' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails. intros [? ->]%envs_lookup_delete_Some ??? HP1 HQ.
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  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.
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  rewrite right_id (into_wand _ R) HP1 assoc -(comm _ P1') -assoc.
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  rewrite -(elim_modal P1' P1 Q Q). apply sep_mono_r, wand_intro_l.
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  by rewrite persistently_if_elim assoc !wand_elim_r.
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Qed.

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Lemma tac_unlock Δ Q : envs_entails Δ Q  envs_entails Δ (locked Q).
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Proof. by unlock. Qed.

Lemma tac_specialize_frame Δ Δ' j q R P1 P2 P1' Q Q' :
  envs_lookup_delete j Δ = Some (q, R, Δ') 
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  IntoWand false R P1 P2 
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  ElimModal P1' P1 Q Q 
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  envs_entails Δ' (P1'  locked Q') 
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  Q' = (P2 - Q)%I 
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  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails. intros [? ->]%envs_lookup_delete_Some ?? HPQ ->.
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  rewrite envs_lookup_sound //. rewrite HPQ -lock.
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  rewrite (into_wand _ R) assoc -(comm _ P1') -assoc persistently_if_elim.
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  rewrite -{2}(elim_modal P1' P1 Q Q). apply sep_mono_r, wand_intro_l.
  by rewrite assoc !wand_elim_r.
Qed.

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Lemma tac_specialize_assert_pure Δ Δ' j q R P1 P2 φ Q :
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  envs_lookup j Δ = Some (q, R) 
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  IntoWand false R P1 P2  FromPure P1 φ 
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  envs_simple_replace j q (Esnoc Enil j P2) Δ = Some Δ' 
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  φ  envs_entails Δ' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails=> ????? <-. rewrite envs_simple_replace_sound //; simpl.
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  rewrite right_id (into_wand _ R) -(from_pure P1) pure_True //.
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  by rewrite wand_True wand_elim_r.
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Qed.

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Lemma tac_specialize_assert_persistent Δ Δ' Δ'' j q P1 P2 R Q :
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  envs_lookup_delete j Δ = Some (q, R, Δ') 
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  IntoWand false R P1 P2  Persistent P1 
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  envs_simple_replace j q (Esnoc Enil j P2) Δ = Some Δ'' 
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  envs_entails Δ' P1  envs_entails Δ'' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails. intros [? ->]%envs_lookup_delete_Some ??? HP1 <-.
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  rewrite envs_lookup_sound //.
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  rewrite -(idemp uPred_and (of_envs (envs_delete _ _ _))).
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  rewrite {1}HP1 (persistent P1) persistently_and_sep_l assoc.
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  rewrite envs_simple_replace_sound' //; simpl.
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  rewrite right_id (into_wand _ R) (persistently_elim_if q) -persistently_if_sep wand_elim_l.
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  by rewrite wand_elim_r.
Qed.

Lemma tac_specialize_persistent_helper Δ Δ' j q P R Q :
  envs_lookup j Δ = Some (q,P) 
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  envs_entails Δ R  Persistent R 
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  envs_replace j q true (Esnoc Enil j R) Δ = Some Δ' 
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  envs_entails Δ' Q  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails. intros ? HR ?? <-.
  rewrite -(idemp uPred_and (of_envs Δ)) {1}HR and_sep_l.
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  rewrite envs_replace_sound //; simpl.
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  by rewrite right_id assoc (sep_elim_l R) persistent_persistently wand_elim_r.
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Qed.

Lemma tac_revert Δ Δ' i p P Q :
  envs_lookup_delete i Δ = Some (p,P,Δ') 
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  envs_entails Δ' ((if p then  P else P)%I - Q) 
  envs_entails Δ Q.
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Proof.