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From iris.bi Require Export bi.
From iris.bi Require Import 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 bi.
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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 (PROP : bi) :=
  Envs { env_persistent : env PROP; env_spatial : env PROP }.
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Add Printing Constructor envs.
Arguments Envs {_} _ _.
Arguments env_persistent {_} _.
Arguments env_spatial {_} _.

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Record envs_wf {PROP} (Δ : envs PROP) := {
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  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 {PROP} (Δ : envs PROP) : PROP :=
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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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Arguments of_envs : simpl never.
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Definition envs_entails {PROP} (Δ : envs PROP) (Q : PROP) : Prop :=
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  of_envs Δ  Q.
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Arguments envs_entails {_} _ _%I : simpl never.
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Typeclasses Opaque envs_entails.
Instance: Params (@envs_entails) 1.

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Record envs_Forall2 {PROP : bi} (R : relation PROP) (Δ1 Δ2 : envs PROP) := {
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  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 {PROP} (Δ : envs PROP) : list ident :=
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  env_dom (env_persistent Δ) ++ env_dom (env_spatial Δ).
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Definition envs_lookup {PROP} (i : ident) (Δ : envs PROP) : option (bool * PROP) :=
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  let (Γp,Γs) := Δ in
  match env_lookup i Γp with
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  | Some P => Some (true, P)
  | None => P  env_lookup i Γs; Some (false, P)
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  end.

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

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Definition envs_lookup_delete {PROP} (i : ident)
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    (Δ : envs PROP) : option (bool * PROP * envs PROP) :=
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  let (Γp,Γs) := Δ in
  match env_lookup_delete i Γp with
  | Some (P,Γp') => Some (true, P, Envs Γp' Γs)
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  | None => ''(P,Γs')  env_lookup_delete i Γs; Some (false, P, Envs Γp Γs')
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  end.

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

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Definition envs_snoc {PROP} (Δ : envs PROP)
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    (p : bool) (j : ident) (P : PROP) : envs PROP :=
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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 {PROP : bi} (p : bool)
    (Γ : env PROP) (Δ : envs PROP) : option (envs PROP) :=
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  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 {PROP : bi} (i : ident) (p : bool)
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    (Γ : env PROP) (Δ : envs PROP) : option (envs PROP) :=
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  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 {PROP : bi} (i : ident) (p q : bool)
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    (Γ : env PROP) (Δ : envs PROP) : option (envs PROP) :=
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  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 {PROP} (Δ : envs PROP) : bool :=
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  if env_spatial Δ is Enil then true else false.

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

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

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Fixpoint envs_split_go {PROP}
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    (js : list ident) (Δ1 Δ2 : envs PROP) : option (envs PROP * envs PROP) :=
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  match js with
  | [] => Some (Δ1, Δ2)
  | j :: js =>
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     ''(p,P,Δ1')  envs_lookup_delete j Δ1;
     if p : bool then envs_split_go js Δ1 Δ2 else
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     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)] *)
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Definition envs_split {PROP} (d : direction)
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    (js : list ident) (Δ : envs PROP) : option (envs PROP * envs PROP) :=
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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 *)
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Section bi_tactics.
Context {PROP : bi}.
Implicit Types Γ : env PROP.
Implicit Types Δ : envs PROP.
Implicit Types P Q : PROP.

Lemma of_envs_eq 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_l=> Hwf.
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  destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
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  - rewrite pure_True ?left_id; last (destruct Hwf; constructor;
      naive_solver eauto using env_delete_wf, env_delete_fresh).
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    rewrite (env_lookup_perm Γp) //= affinely_persistently_and.
    by rewrite and_sep_affinely_persistently -assoc.
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  - destruct (Γs !! i) eqn:?; simplify_eq/=.
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    rewrite pure_True ?left_id; last (destruct Hwf; constructor;
      naive_solver eauto using env_delete_wf, env_delete_fresh).
    rewrite (env_lookup_perm Γs) //=. by rewrite !assoc -(comm _ P).
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. rewrite -persistently_and_affinely_sep_l. apply and_intro; last done.
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  rewrite envs_lookup_sound //; simpl.
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  by rewrite -persistently_and_affinely_sep_l and_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_l=> Hwf.
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  destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
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  - rewrite pure_True // left_id (env_lookup_perm Γp) //=
            affinely_persistently_and and_sep_affinely_persistently.
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    cancel [ P]%I. apply wand_intro_l. solve_sep_entails.
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  - 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.

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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 affinely_persistently_emp 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 -affinely_persistently_if_sep_2 -assoc. destruct q1; simpl.
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  - destruct rp.
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    + rewrite envs_lookup_sound //; simpl.
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      by rewrite IH // (affinely_persistently_affinely_persistently_if q2).
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    + rewrite envs_lookup_persistent_sound //.
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      by rewrite IH // (affinely_persistently_affinely_persistently_if q2).
  - rewrite envs_lookup_sound // IH //; simpl. by rewrite affinely_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.
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  rewrite /envs_lookup /envs_snoc /of_envs=> ?; apply pure_elim_l=> Hwf.
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  destruct Δ as [Γp Γs], (Γp !! i) eqn:?, (Γs !! i) eqn:?; simplify_eq/=.
  apply wand_intro_l; destruct p; simpl.
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  - apply and_intro; [apply pure_intro|].
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    + 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 affinely_persistently_and and_sep_affinely_persistently assoc.
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  - apply and_intro; [apply pure_intro|].
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    + 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 :
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  envs_app p Γ Δ = Some Δ' 
  of_envs Δ  (if p then  [] Γ else [] Γ) - of_envs Δ'.
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Proof.
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  rewrite /of_envs /envs_app=> ?; apply pure_elim_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, and_intro; [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 affinely_persistently_and and_sep_affinely_persistently.
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      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, and_intro; [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.

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Lemma envs_app_singleton_sound Δ Δ' p j Q :
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  envs_app p (Esnoc Enil j Q) Δ = Some Δ'  of_envs Δ  ?p Q - of_envs Δ'.
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Proof. move=> /envs_app_sound. destruct p; by rewrite /= right_id. Qed.

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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)  (if p then  [] Γ else [] Γ) - of_envs Δ'.
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Proof.
  rewrite /envs_simple_replace /envs_delete /of_envs=> ?.
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  apply pure_elim_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, and_intro; [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 affinely_persistently_and and_sep_affinely_persistently.
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      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, and_intro; [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.

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Lemma envs_simple_replace_singleton_sound' Δ Δ' i p j Q :
  envs_simple_replace i p (Esnoc Enil j Q) Δ = Some Δ' 
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  of_envs (envs_delete i p Δ)  ?p Q - of_envs Δ'.
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Proof. move=> /envs_simple_replace_sound'. destruct p; by rewrite /= right_id. Qed.

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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  ((if p then  [] Γ else [] Γ) - of_envs Δ').
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Proof. intros. by rewrite envs_lookup_sound// envs_simple_replace_sound'//. Qed.

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Lemma envs_simple_replace_singleton_sound Δ Δ' i p P j Q :
  envs_lookup i Δ = Some (p,P) 
  envs_simple_replace i p (Esnoc Enil j Q) Δ = Some Δ' 
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  of_envs Δ  ?p P  (?p Q - of_envs Δ').
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Proof.
  intros. by rewrite envs_lookup_sound// envs_simple_replace_singleton_sound'//.
Qed.

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Lemma envs_replace_sound' Δ Δ' i p q Γ :
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  envs_replace i p q Γ Δ = Some Δ' 
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  of_envs (envs_delete i p Δ)  (if q then  [] Γ else [] 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.

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Lemma envs_replace_singleton_sound' Δ Δ' i p q j Q :
  envs_replace i p q (Esnoc Enil j Q) Δ = Some Δ' 
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  of_envs (envs_delete i p Δ)  ?q Q - of_envs Δ'.
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Proof. move=> /envs_replace_sound'. destruct q; by rewrite /= ?right_id. Qed.

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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  ((if q then  [] Γ else [] 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_replace_singleton_sound Δ Δ' i p q P j Q :
  envs_lookup i Δ = Some (p,P) 
  envs_replace i p q (Esnoc Enil j Q) Δ = Some Δ' 
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  of_envs Δ  ?p P  (?q Q - of_envs Δ').
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Proof. intros. by rewrite envs_lookup_sound// envs_replace_singleton_sound'//. Qed.

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Lemma envs_lookup_envs_clear_spatial Δ j :
  envs_lookup j (envs_clear_spatial Δ)
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  = ''(p,P)  envs_lookup j Δ; if p : bool 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 :
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  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_l=> Hwf.
  rewrite right_id -persistent_and_sep_assoc. apply and_intro; [|done].
  apply pure_intro. destruct Hwf; constructor; simpl; auto using Enil_wf.
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Qed.

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Lemma env_spatial_is_nil_affinely_persistently Δ :
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  env_spatial_is_nil Δ = true  of_envs Δ   of_envs Δ.
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Proof.
  intros. unfold of_envs; destruct Δ as [? []]; simplify_eq/=.
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  rewrite !right_id {1}affinely_and_r persistently_and.
  by rewrite persistently_affinely persistently_idemp persistently_pure.
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Qed.
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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') 
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  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|].
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  apply pure_elim with (envs_wf Δ1)=> [|Hwf].
  { by rewrite /of_envs !and_elim_l sep_elim_l. }
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  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 bi_and (of_envs Δ)).
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  rewrite {2}envs_clear_spatial_sound.
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  rewrite (env_spatial_is_nil_affinely_persistently (envs_clear_spatial _)) //.
  rewrite -persistently_and_affinely_sep_l.
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  rewrite (and_elim_l (bi_persistently _)%I)
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          persistently_and_affinely_sep_r affinely_persistently_elim.
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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 PROP) :
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  Reflexive R  Reflexive (envs_Forall2 R).
Proof. by constructor. Qed.
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Global Instance envs_Forall2_sym (R : relation PROP) :
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  Symmetric R  Symmetric (envs_Forall2 R).
Proof. intros ??? [??]; by constructor. Qed.
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Global Instance envs_Forall2_trans (R : relation PROP) :
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  Transitive R  Transitive (envs_Forall2 R).
Proof. intros ??? [??] [??] [??]; constructor; etrans; eauto. Qed.
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Global Instance envs_Forall2_antisymm (R R' : relation PROP) :
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  AntiSymm R R'  AntiSymm (envs_Forall2 R) (envs_Forall2 R').
Proof. intros ??? [??] [??]; constructor; by eapply (anti_symm _). Qed.
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Lemma envs_Forall2_impl (R R' : relation PROP) Δ1 Δ2 :
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  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.

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Global Instance of_envs_mono : Proper (envs_Forall2 () ==> ()) (@of_envs PROP).
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Proof.
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  intros [Γp1 Γs1] [Γp2 Γs2] [Hp Hs]; apply and_mono; simpl in *.
  - apply pure_mono=> -[???]. constructor;
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      naive_solver eauto using env_Forall2_wf, env_Forall2_fresh.
  - by repeat f_equiv.
Qed.
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Global Instance of_envs_proper :
  Proper (envs_Forall2 () ==> ()) (@of_envs PROP).
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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 PROP) :
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  Proper (env_Forall2 R ==> env_Forall2 R ==> envs_Forall2 R) (@Envs PROP).
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Proof. by constructor. Qed.

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Global Instance envs_entails_proper :
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  Proper (envs_Forall2 () ==> () ==> iff) (@envs_entails PROP).
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Proof. solve_proper. Qed.
Global Instance envs_entails_flip_mono :
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  Proper (envs_Forall2 () ==> flip () ==> flip impl) (@envs_entails PROP).
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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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  intros <-. rewrite /of_envs /= persistently_True_emp affinely_persistently_emp left_id.
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  apply and_intro=> //. apply pure_intro; repeat constructor.
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Qed.

(** * Basic rules *)
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Lemma tac_eval Δ Q Q' :
  Q = Q' 
  envs_entails Δ Q'  envs_entails Δ Q.
Proof. by intros ->. Qed.

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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.

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(* If the BI is affine, no need to walk on the whole environment. *)
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Global Instance affine_env_bi `(BiAffine PROP) Γ : AffineEnv Γ | 0.
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Proof. induction Γ; apply _. Qed.

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

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Lemma tac_emp_intro Δ : AffineEnv (env_spatial Δ)  of_envs Δ  emp.
Proof. intros. by rewrite (affine (of_envs Δ)). Qed.
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Lemma tac_assumption Δ Δ' i p P Q :
  envs_lookup_delete i Δ = Some (p,P,Δ') 
  FromAssumption p P Q 
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  (if env_spatial_is_nil Δ' then TCTrue
   else TCOr (Absorbing Q) (AffineEnv (env_spatial Δ'))) 
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  of_envs Δ  Q.
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Proof.
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  intros ?? H. rewrite envs_lookup_delete_sound //.
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  destruct (env_spatial_is_nil Δ') eqn:?.
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  - by rewrite (env_spatial_is_nil_affinely_persistently Δ') // sep_elim_l.
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  - rewrite from_assumption. destruct H; by rewrite sep_elim_l.
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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_singleton_sound //.
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  by rewrite wand_elim_r.
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Qed.
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Lemma tac_clear Δ Δ' i p P Q :
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  envs_lookup_delete i Δ = Some (p,P,Δ') 
  (if p then TCTrue else TCOr (Affine P) (Absorbing Q)) 
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  envs_entails Δ' Q 
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  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails=> ?? HQ. rewrite envs_lookup_delete_sound //.
  by destruct p; rewrite /= HQ sep_elim_r.
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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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Lemma tac_false_destruct Δ i p P Q :
  envs_lookup i Δ = Some (p,P) 
  P = False%I 
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  envs_entails Δ Q.
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Proof.
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  rewrite /envs_entails => ??. subst. rewrite envs_lookup_sound //; simpl.
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  by rewrite affinely_persistently_if_elim sep_elim_l False_elim.
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Qed.

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(** * Pure *)
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Lemma tac_pure_intro Δ Q φ : FromPure false 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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  (if p then TCTrue else TCOr (Affine P) (Absorbing Q)) 
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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=> ?? HPQ HQ.
  rewrite envs_lookup_delete_sound //; simpl. destruct p; simpl.
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  - rewrite (into_pure P) -persistently_and_affinely_sep_l persistently_pure.
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    by apply pure_elim_l.
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  - destruct HPQ.
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    + rewrite -(affine_affinely P) (into_pure P) -persistent_and_affinely_sep_l.
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      by apply pure_elim_l.
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    + rewrite (into_pure P) (persistent_absorbingly_affinely  _ %I) absorbingly_sep_lr.
      rewrite -persistent_and_affinely_sep_l. apply pure_elim_l=> ?. by rewrite HQ.
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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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(** * Always modalities *)
Class FilterPersistentEnv
    (M : always_modality PROP) (C : PROP  Prop) (Γ1 Γ2 : env PROP) := {
  filter_persistent_env :
    ( P, C P   P  M ( P)) 
     ([] Γ1)  M ( ([] Γ2));
  filter_persistent_env_wf : env_wf Γ1  env_wf Γ2;
  filter_persistent_env_dom i : Γ1 !! i = None  Γ2 !! i = None;
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}.
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Global Instance filter_persistent_env_nil M C : FilterPersistentEnv M C Enil Enil.
Proof.
  split=> // HC /=. rewrite !persistently_pure !affinely_True_emp.
  by rewrite affinely_emp -always_modality_emp.
Qed.
Global Instance filter_persistent_env_snoc M (C : PROP  Prop) Γ Γ' i P :
  C P 
  FilterPersistentEnv M C Γ Γ' 
  FilterPersistentEnv M C (Esnoc Γ i P) (Esnoc Γ' i P).
Proof.
  intros ? [HΓ Hwf Hdom]; split; simpl.
  - intros HC. rewrite affinely_persistently_and HC // HΓ //.
    by rewrite always_modality_and -affinely_persistently_and.
  - inversion 1; constructor; auto.
  - intros j. destruct (ident_beq _ _); naive_solver.
Qed.
Global Instance filter_persistent_env_snoc_not M (C : PROP  Prop) Γ Γ' i P :
  FilterPersistentEnv M C Γ Γ' 
  FilterPersistentEnv M C (Esnoc Γ i P) Γ' | 100.
Proof.
  intros [HΓ Hwf Hdom]; split; simpl.
  - intros HC. by rewrite and_elim_r HΓ.
  - inversion 1; auto.
  - intros j. destruct (ident_beq _ _); naive_solver.
Qed.
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Class FilterSpatialEnv
    (M : always_modality PROP) (C : PROP  Prop) (Γ1 Γ2 : env PROP) := {
  filter_spatial_env :
    ( P, C P  P  M P)  ( P, Absorbing (M P)) 
    ([] Γ1)  M ([] Γ2);
  filter_spatial_env_wf : env_wf Γ1  env_wf Γ2;
  filter_spatial_env_dom i : Γ1 !! i = None  Γ2 !! i = None;
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}.
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Global Instance filter_spatial_env_nil M C : FilterSpatialEnv M C Enil Enil.
Proof. split=> // HC /=. by rewrite -always_modality_emp. Qed.
Global Instance filter_spatial_env_snoc M (C : PROP  Prop) Γ Γ' i P :
  C P 
  FilterSpatialEnv M C Γ Γ' 
  FilterSpatialEnv M C (Esnoc Γ i P) (Esnoc Γ' i P).
Proof.
  intros ? [HΓ Hwf Hdom]; split; simpl.
  - intros HC ?. by rewrite {1}(HC P) // HΓ // always_modality_sep.
  - inversion 1; constructor; auto.
  - intros j. destruct (ident_beq _ _); naive_solver.
Qed.

Global Instance filter_spatial_env_snoc_not M (C : PROP  Prop) Γ Γ' i P :
  FilterSpatialEnv M C Γ Γ' 
  FilterSpatialEnv M C (Esnoc Γ i P) Γ' | 100.
Proof.
  intros [HΓ Hwf Hdom]; split; simpl.
  - intros HC ?. by rewrite HΓ // sep_elim_r.
  - inversion 1; auto.
  - intros j. destruct (ident_beq _ _); naive_solver.
Qed.
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Ltac tac_always_cases :=
  simplify_eq/=;
  repeat match goal with
  | H : TCAnd _ _ |- _ => destruct H
  | H : TCEq ?x _ |- _ => inversion H; subst x; clear H
  | H : TCForall _ _ |- _ => apply TCForall_Forall in H
  | H : FilterPersistentEnv _ _ _ _ |- _ => destruct H
  | H : FilterSpatialEnv _ _ _ _ |- _ => destruct H
  end; simpl; auto using Enil_wf.

Lemma tac_always_intro Γp Γs Γp' Γs' M Q Q' :
  FromAlways M Q' Q 
  match always_modality_persistent_spec M with
  | AIEnvForall C => TCAnd (TCForall C (env_to_list Γp)) (TCEq Γp Γp')
  | AIEnvFilter C => FilterPersistentEnv M C Γp Γp'
  | AIEnvIsEmpty => TCAnd (TCEq Γp Enil) (TCEq Γp' Enil)
  | AIEnvClear => TCEq Γp' Enil
  | AIEnvId => TCEq Γp Γp'
  end 
  match always_modality_spatial_spec M with
  | AIEnvForall C => TCAnd (TCForall C (env_to_list Γs)) (TCEq Γs Γs')
  | AIEnvFilter C => FilterSpatialEnv M C Γs Γs'
  | AIEnvIsEmpty => TCAnd (TCEq Γs Enil) (TCEq Γs' Enil)
  | AIEnvClear => TCEq Γs' Enil
  | AIEnvId => TCEq Γs Γs'
  end 
  envs_entails (Envs Γp' Γs') Q  envs_entails (Envs Γp Γs) Q'.
Proof.
  rewrite /envs_entails /FromAlways /of_envs /= => <- HΓp HΓs <-.
  apply pure_elim_l=> -[???]. assert (envs_wf (Envs Γp' Γs')).
  { split; simpl in *.
    - destruct (always_modality_persistent_spec M); tac_always_cases.
    - destruct (always_modality_spatial_spec M); tac_always_cases.
    - destruct (always_modality_persistent_spec M),
        (always_modality_spatial_spec M); tac_always_cases; naive_solver. }
  rewrite pure_True // left_id. rewrite -always_modality_sep. apply sep_mono.
  - destruct (always_modality_persistent_spec M) eqn:?; tac_always_cases.
    + by rewrite {1}affinely_elim_emp (always_modality_emp M)
        persistently_True_emp affinely_persistently_emp.
    + eauto using always_modality_persistent_forall_big_and.
    + eauto using always_modality_persistent_filter.
    + by rewrite {1}affinely_elim_emp (always_modality_emp M)
        persistently_True_emp affinely_persistently_emp.
    + eauto using always_modality_persistent_id.
  - destruct (always_modality_spatial_spec M) eqn:?; tac_always_cases.
    + by rewrite -always_modality_emp.
    + eauto using always_modality_spatial_forall_big_sep.
    + eauto using always_modality_spatial_filter,
        always_modality_spatial_filter_absorbing.
    + rewrite -(always_modality_spatial_clear M) // -always_modality_emp.
      by rewrite -absorbingly_True_emp absorbingly_pure -True_intro.
    + by destruct (always_modality_spatial_id M).
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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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  (if p then TCTrue else TCOr (Affine P) (Absorbing Q)) 
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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=>?? HPQ ? HQ. rewrite envs_replace_singleton_sound //=.
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  destruct p; simpl.
  - by rewrite -(into_persistent _ P) /= wand_elim_r.
  - destruct HPQ.
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    + rewrite -(affine_affinely P) (_ : P = bi_persistently_if false P)%I //
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              (into_persistent _ P) wand_elim_r //.
    + rewrite (_ : P = bi_persistently_if false P)%I // (into_persistent _ P).
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      by rewrite {1}(persistent_absorbingly_affinely (bi_persistently _)%I)
                 absorbingly_sep_l wand_elim_r HQ.
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Qed.

(** * Implication and wand *)
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Lemma envs_app_singleton_sound_foo Δ Δ' p j Q :
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  envs_app p (Esnoc Enil j Q) Δ = Some Δ'  of_envs Δ  ?p Q  of_envs Δ'.
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Proof. intros. apply wand_elim_l'. eapply envs_app_singleton_sound. eauto. Qed.

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Lemma tac_impl_intro Δ Δ' i P P' Q R :
  FromImpl R 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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  FromAffinely P' P 
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  envs_entails Δ' Q  envs_entails Δ R.
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Proof.
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  rewrite /FromImpl /envs_entails => <- ??? <-. destruct (env_spatial_is_nil Δ) eqn:?.
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  - rewrite (env_spatial_is_nil_affinely_persistently Δ) //; simpl. apply impl_intro_l.
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    rewrite envs_app_singleton_sound //; simpl.
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    rewrite -(from_affinely P') -affinely_and_lr.
    by rewrite persistently_and_affinely_sep_r affinely_persistently_elim wand_elim_r.
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  - apply impl_intro_l. rewrite envs_app_singleton_sound //; simpl.
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    by rewrite -(from_affinely P') persistent_and_affinely_sep_l_1 wand_elim_r.
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703
Qed.
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Lemma tac_impl_intro_persistent Δ Δ' i P P' Q R :
  FromImpl R P Q 
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  IntoPersistent false P P' 
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