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From iris.bi Require Export derived_connectives.
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From iris.algebra Require Import monoid.

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(** Naming schema for lemmas about modalities:
    M1_into_M2: M1 P ⊢ M2 P
    M1_M2_elim: M1 (M2 P) ⊣⊢ M1 P
    M1_elim_M2: M1 (M2 P) ⊣⊢ M2 P
    M1_M2: M1 (M2 P) ⊣⊢ M2 (M1 P)
*)

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Module bi.
Import interface.bi.
Section bi_derived.
Context {PROP : bi}.
Implicit Types φ : Prop.
Implicit Types P Q R : PROP.
Implicit Types Ps : list PROP.
Implicit Types A : Type.

Hint Extern 100 (NonExpansive _) => solve_proper.

(* Force implicit argument PROP *)
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Notation "P ⊢ Q" := (P @{PROP} Q).
Notation "P ⊣⊢ Q" := (P @{PROP} Q).
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(* Derived stuff about the entailment *)
Global Instance entails_anti_sym : AntiSymm () (@bi_entails PROP).
Proof. intros P Q ??. by apply equiv_spec. Qed.
Lemma equiv_entails P Q : (P  Q)  (P  Q).
Proof. apply equiv_spec. Qed.
Lemma equiv_entails_sym P Q : (Q  P)  (P  Q).
Proof. apply equiv_spec. Qed.
Global Instance entails_proper :
  Proper (() ==> () ==> iff) (() : relation PROP).
Proof.
  move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split=>?.
  - by trans P1; [|trans Q1].
  - by trans P2; [|trans Q2].
Qed.
Lemma entails_equiv_l P Q R : (P  Q)  (Q  R)  (P  R).
Proof. by intros ->. Qed.
Lemma entails_equiv_r P Q R : (P  Q)  (Q  R)  (P  R).
Proof. by intros ? <-. Qed.
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Global Instance bi_emp_valid_proper : Proper (() ==> iff) (@bi_emp_valid PROP).
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Proof. solve_proper. Qed.
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Global Instance bi_emp_valid_mono : Proper (() ==> impl) (@bi_emp_valid PROP).
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Proof. solve_proper. Qed.
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Global Instance bi_emp_valid_flip_mono :
  Proper (flip () ==> flip impl) (@bi_emp_valid PROP).
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Proof. solve_proper. Qed.

(* Propers *)
Global Instance pure_proper : Proper (iff ==> ()) (@bi_pure PROP) | 0.
Proof. intros φ1 φ2 Hφ. apply equiv_dist=> n. by apply pure_ne. Qed.
Global Instance and_proper :
  Proper (() ==> () ==> ()) (@bi_and PROP) := ne_proper_2 _.
Global Instance or_proper :
  Proper (() ==> () ==> ()) (@bi_or PROP) := ne_proper_2 _.
Global Instance impl_proper :
  Proper (() ==> () ==> ()) (@bi_impl PROP) := ne_proper_2 _.
Global Instance sep_proper :
  Proper (() ==> () ==> ()) (@bi_sep PROP) := ne_proper_2 _.
Global Instance wand_proper :
  Proper (() ==> () ==> ()) (@bi_wand PROP) := ne_proper_2 _.
Global Instance forall_proper A :
  Proper (pointwise_relation _ () ==> ()) (@bi_forall PROP A).
Proof.
  intros Φ1 Φ2 HΦ. apply equiv_dist=> n.
  apply forall_ne=> x. apply equiv_dist, HΦ.
Qed.
Global Instance exist_proper A :
  Proper (pointwise_relation _ () ==> ()) (@bi_exist PROP A).
Proof.
  intros Φ1 Φ2 HΦ. apply equiv_dist=> n.
  apply exist_ne=> x. apply equiv_dist, HΦ.
Qed.
Global Instance persistently_proper :
  Proper (() ==> ()) (@bi_persistently PROP) := ne_proper _.

(* Derived logical stuff *)
Lemma and_elim_l' P Q R : (P  R)  P  Q  R.
Proof. by rewrite and_elim_l. Qed.
Lemma and_elim_r' P Q R : (Q  R)  P  Q  R.
Proof. by rewrite and_elim_r. Qed.
Lemma or_intro_l' P Q R : (P  Q)  P  Q  R.
Proof. intros ->; apply or_intro_l. Qed.
Lemma or_intro_r' P Q R : (P  R)  P  Q  R.
Proof. intros ->; apply or_intro_r. Qed.
Lemma exist_intro' {A} P (Ψ : A  PROP) a : (P  Ψ a)  P   a, Ψ a.
Proof. intros ->; apply exist_intro. Qed.
Lemma forall_elim' {A} P (Ψ : A  PROP) : (P   a, Ψ a)   a, P  Ψ a.
Proof. move=> HP a. by rewrite HP forall_elim. Qed.

Hint Resolve pure_intro forall_intro.
Hint Resolve or_elim or_intro_l' or_intro_r'.
Hint Resolve and_intro and_elim_l' and_elim_r'.

Lemma impl_intro_l P Q R : (Q  P  R)  P  Q  R.
Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed.
Lemma impl_elim P Q R : (P  Q  R)  (P  Q)  P  R.
Proof. intros. rewrite -(impl_elim_l' P Q R); auto. Qed.
Lemma impl_elim_r' P Q R : (Q  P  R)  P  Q  R.
Proof. intros; apply impl_elim with P; auto. Qed.
Lemma impl_elim_l P Q : (P  Q)  P  Q.
Proof. by apply impl_elim_l'. Qed.
Lemma impl_elim_r P Q : P  (P  Q)  Q.
Proof. by apply impl_elim_r'. Qed.

Lemma False_elim P : False  P.
Proof. by apply (pure_elim' False). Qed.
Lemma True_intro P : P  True.
Proof. by apply pure_intro. Qed.
Hint Immediate False_elim.

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Lemma entails_eq_True P Q : (P  Q)  ((P  Q)%I  True%I).
Proof.
  split=>EQ.
  - apply bi.equiv_spec; split; [by apply True_intro|].
    apply impl_intro_r. rewrite and_elim_r //.
  - trans (P  True)%I.
    + apply and_intro; first done. by apply pure_intro.
    + rewrite -EQ impl_elim_r. done.
Qed.
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Lemma entails_impl_True P Q : (P  Q)  (True  (P  Q)).
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Proof. rewrite entails_eq_True equiv_spec; naive_solver. Qed.
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Lemma and_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q'.
Proof. auto. Qed.
Lemma and_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
Proof. by intros; apply and_mono. Qed.
Lemma and_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
Proof. by apply and_mono. Qed.

Lemma or_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q'.
Proof. auto. Qed.
Lemma or_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
Proof. by intros; apply or_mono. Qed.
Lemma or_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
Proof. by apply or_mono. Qed.

Lemma impl_mono P P' Q Q' : (Q  P)  (P'  Q')  (P  P')  Q  Q'.
Proof.
  intros HP HQ'; apply impl_intro_l; rewrite -HQ'.
  apply impl_elim with P; eauto.
Qed.
Lemma forall_mono {A} (Φ Ψ : A  PROP) :
  ( a, Φ a  Ψ a)  ( a, Φ a)   a, Ψ a.
Proof.
  intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim.
Qed.
Lemma exist_mono {A} (Φ Ψ : A  PROP) :
  ( a, Φ a  Ψ a)  ( a, Φ a)   a, Ψ a.
Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed.

Global Instance and_mono' : Proper (() ==> () ==> ()) (@bi_and PROP).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance and_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@bi_and PROP).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance or_mono' : Proper (() ==> () ==> ()) (@bi_or PROP).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance or_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@bi_or PROP).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance impl_mono' :
  Proper (flip () ==> () ==> ()) (@bi_impl PROP).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
Global Instance impl_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@bi_impl PROP).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
Global Instance forall_mono' A :
  Proper (pointwise_relation _ () ==> ()) (@bi_forall PROP A).
Proof. intros P1 P2; apply forall_mono. Qed.
Global Instance forall_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@bi_forall PROP A).
Proof. intros P1 P2; apply forall_mono. Qed.
Global Instance exist_mono' A :
  Proper (pointwise_relation _ (()) ==> ()) (@bi_exist PROP A).
Proof. intros P1 P2; apply exist_mono. Qed.
Global Instance exist_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@bi_exist PROP A).
Proof. intros P1 P2; apply exist_mono. Qed.

Global Instance and_idem : IdemP () (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_idem : IdemP () (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_comm : Comm () (@bi_and PROP).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance True_and : LeftId () True%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_True : RightId () True%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_and : LeftAbsorb () False%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_False : RightAbsorb () False%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance True_or : LeftAbsorb () True%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_True : RightAbsorb () True%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_or : LeftId () False%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_False : RightId () False%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_assoc : Assoc () (@bi_and PROP).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance or_comm : Comm () (@bi_or PROP).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance or_assoc : Assoc () (@bi_or PROP).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance True_impl : LeftId () True%I (@bi_impl PROP).
Proof.
  intros P; apply (anti_symm ()).
  - by rewrite -(left_id True%I ()%I (_  _)%I) impl_elim_r.
  - by apply impl_intro_l; rewrite left_id.
Qed.

Lemma False_impl P : (False  P)  True.
Proof.
  apply (anti_symm ()); [by auto|].
  apply impl_intro_l. rewrite left_absorb. auto.
Qed.

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Lemma exist_impl_forall {A} P (Ψ : A  PROP) :
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  (( x : A, Ψ x)  P)   x : A, Ψ x  P.
Proof.
  apply equiv_spec; split.
  - apply forall_intro=>x. by rewrite -exist_intro.
  - apply impl_intro_r, impl_elim_r', exist_elim=>x.
    apply impl_intro_r. by rewrite (forall_elim x) impl_elim_r.
Qed.
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Lemma forall_unit (Ψ : unit  PROP) :
  ( x, Ψ x)  Ψ ().
Proof.
  apply (anti_symm ()).
  - rewrite (forall_elim ()) //.
  - apply forall_intro=>[[]]. done.
Qed.
Lemma exist_unit (Ψ : unit  PROP) :
  ( x, Ψ x)  Ψ ().
Proof.
  apply (anti_symm ()).
  - apply exist_elim=>[[]]. done.
  - rewrite -(exist_intro ()). done.
Qed.
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Lemma or_and_l P Q R : P  Q  R  (P  Q)  (P  R).
Proof.
  apply (anti_symm ()); first auto.
  do 2 (apply impl_elim_l', or_elim; apply impl_intro_l); auto.
Qed.
Lemma or_and_r P Q R : P  Q  R  (P  R)  (Q  R).
Proof. by rewrite -!(comm _ R) or_and_l. Qed.
Lemma and_or_l P Q R : P  (Q  R)  P  Q  P  R.
Proof.
  apply (anti_symm ()); last auto.
  apply impl_elim_r', or_elim; apply impl_intro_l; auto.
Qed.
Lemma and_or_r P Q R : (P  Q)  R  P  R  Q  R.
Proof. by rewrite -!(comm _ R) and_or_l. Qed.
Lemma and_exist_l {A} P (Ψ : A  PROP) : P  ( a, Ψ a)   a, P  Ψ a.
Proof.
  apply (anti_symm ()).
  - apply impl_elim_r'. apply exist_elim=>a. apply impl_intro_l.
    by rewrite -(exist_intro a).
  - apply exist_elim=>a. apply and_intro; first by rewrite and_elim_l.
    by rewrite -(exist_intro a) and_elim_r.
Qed.
Lemma and_exist_r {A} P (Φ: A  PROP) : ( a, Φ a)  P   a, Φ a  P.
Proof.
  rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm.
Qed.
Lemma or_exist {A} (Φ Ψ : A  PROP) :
  ( a, Φ a  Ψ a)  ( a, Φ a)  ( a, Ψ a).
Proof.
  apply (anti_symm ()).
  - apply exist_elim=> a. by rewrite -!(exist_intro a).
  - apply or_elim; apply exist_elim=> a; rewrite -(exist_intro a); auto.
Qed.

Lemma and_alt P Q : P  Q   b : bool, if b then P else Q.
Proof.
   apply (anti_symm _); first apply forall_intro=> -[]; auto.
   by apply and_intro; [rewrite (forall_elim true)|rewrite (forall_elim false)].
Qed.
Lemma or_alt P Q : P  Q   b : bool, if b then P else Q.
Proof.
  apply (anti_symm _); last apply exist_elim=> -[]; auto.
  by apply or_elim; [rewrite -(exist_intro true)|rewrite -(exist_intro false)].
Qed.

Lemma entails_equiv_and P Q : (P  Q  P)  (P  Q).
Proof. split. by intros ->; auto. intros; apply (anti_symm _); auto. Qed.

Global Instance iff_ne : NonExpansive2 (@bi_iff PROP).
Proof. unfold bi_iff; solve_proper. Qed.
Global Instance iff_proper :
  Proper (() ==> () ==> ()) (@bi_iff PROP) := ne_proper_2 _.

Lemma iff_refl Q P : Q  P  P.
Proof. rewrite /bi_iff; apply and_intro; apply impl_intro_l; auto. Qed.


(* BI Stuff *)
Hint Resolve sep_mono.
Lemma sep_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
Proof. by intros; apply sep_mono. Qed.
Lemma sep_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
Proof. by apply sep_mono. Qed.
Global Instance sep_mono' : Proper (() ==> () ==> ()) (@bi_sep PROP).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Global Instance sep_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@bi_sep PROP).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Lemma wand_mono P P' Q Q' : (Q  P)  (P'  Q')  (P - P')  Q - Q'.
Proof.
  intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'.
Qed.
Global Instance wand_mono' : Proper (flip () ==> () ==> ()) (@bi_wand PROP).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
Global Instance wand_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@bi_wand PROP).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.

Global Instance sep_comm : Comm () (@bi_sep PROP).
Proof. intros P Q; apply (anti_symm _); auto using sep_comm'. Qed.
Global Instance sep_assoc : Assoc () (@bi_sep PROP).
Proof.
  intros P Q R; apply (anti_symm _); auto using sep_assoc'.
  by rewrite !(comm _ P) !(comm _ _ R) sep_assoc'.
Qed.
Global Instance emp_sep : LeftId () emp%I (@bi_sep PROP).
Proof. intros P; apply (anti_symm _); auto using emp_sep_1, emp_sep_2. Qed.
Global Instance sep_emp : RightId () emp%I (@bi_sep PROP).
Proof. by intros P; rewrite comm left_id. Qed.

Global Instance sep_False : LeftAbsorb () False%I (@bi_sep PROP).
Proof. intros P; apply (anti_symm _); auto using wand_elim_l'. Qed.
Global Instance False_sep : RightAbsorb () False%I (@bi_sep PROP).
Proof. intros P. by rewrite comm left_absorb. Qed.

Lemma True_sep_2 P : P  True  P.
Proof. rewrite -{1}[P](left_id emp%I bi_sep). auto using sep_mono. Qed.
Lemma sep_True_2 P : P  P  True.
Proof. by rewrite comm -True_sep_2. Qed.

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Lemma sep_intro_emp_valid_l P Q R : P  (R  Q)  R  P  Q.
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Proof. intros ? ->. rewrite -{1}(left_id emp%I _ Q). by apply sep_mono. Qed.
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Lemma sep_intro_emp_valid_r P Q R : (R  P)  Q  R  P  Q.
Proof. intros -> ?. rewrite comm. by apply sep_intro_emp_valid_l. Qed.
Lemma sep_elim_emp_valid_l P Q R : P  (P  R  Q)  R  Q.
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Proof. intros <- <-. by rewrite left_id. Qed.
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Lemma sep_elim_emp_valid_r P Q R : P  (R  P  Q)  R  Q.
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Proof. intros <- <-. by rewrite right_id. Qed.

Lemma wand_intro_l P Q R : (Q  P  R)  P  Q - R.
Proof. rewrite comm; apply wand_intro_r. Qed.
Lemma wand_elim_l P Q : (P - Q)  P  Q.
Proof. by apply wand_elim_l'. Qed.
Lemma wand_elim_r P Q : P  (P - Q)  Q.
Proof. rewrite (comm _ P); apply wand_elim_l. Qed.
Lemma wand_elim_r' P Q R : (Q  P - R)  P  Q  R.
Proof. intros ->; apply wand_elim_r. Qed.
Lemma wand_apply P Q R S : (P  Q - R)  (S  P  Q)  S  R.
Proof. intros HR%wand_elim_l' HQ. by rewrite HQ. Qed.
Lemma wand_frame_l P Q R : (Q - R)  P  Q - P  R.
Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed.
Lemma wand_frame_r P Q R : (Q - R)  Q  P - R  P.
Proof.
  apply wand_intro_l. rewrite ![(_  P)%I]comm -assoc.
  apply sep_mono_r, wand_elim_r.
Qed.

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Global Instance emp_wand : LeftId () emp%I (@bi_wand PROP).
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Proof.
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  intros P. apply (anti_symm _).
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  - by rewrite -[(emp - P)%I]left_id wand_elim_r.
  - apply wand_intro_l. by rewrite left_id.
Qed.
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Lemma False_wand P : (False - P)  True.
Proof.
  apply (anti_symm ()); [by auto|].
  apply wand_intro_l. rewrite left_absorb. auto.
Qed.

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Lemma wand_trans P Q R : (P - Q)  (Q - R)  (P - R).
Proof. apply wand_intro_l. by rewrite assoc !wand_elim_r. Qed.

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Lemma wand_curry P Q R : (P - Q - R)  (P  Q - R).
Proof.
  apply (anti_symm _).
  - apply wand_intro_l. by rewrite (comm _ P) -assoc !wand_elim_r.
  - do 2 apply wand_intro_l. by rewrite assoc (comm _ Q) wand_elim_r.
Qed.

Lemma sep_and_l P Q R : P  (Q  R)  (P  Q)  (P  R).
Proof. auto. Qed.
Lemma sep_and_r P Q R : (P  Q)  R  (P  R)  (Q  R).
Proof. auto. Qed.
Lemma sep_or_l P Q R : P  (Q  R)  (P  Q)  (P  R).
Proof.
  apply (anti_symm ()); last by eauto 8.
  apply wand_elim_r', or_elim; apply wand_intro_l; auto.
Qed.
Lemma sep_or_r P Q R : (P  Q)  R  (P  R)  (Q  R).
Proof. by rewrite -!(comm _ R) sep_or_l. Qed.
Lemma sep_exist_l {A} P (Ψ : A  PROP) : P  ( a, Ψ a)   a, P  Ψ a.
Proof.
  intros; apply (anti_symm ()).
  - apply wand_elim_r', exist_elim=>a. apply wand_intro_l.
    by rewrite -(exist_intro a).
  - apply exist_elim=> a; apply sep_mono; auto using exist_intro.
Qed.
Lemma sep_exist_r {A} (Φ: A  PROP) Q: ( a, Φ a)  Q   a, Φ a  Q.
Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed.
Lemma sep_forall_l {A} P (Ψ : A  PROP) : P  ( a, Ψ a)   a, P  Ψ a.
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.
Lemma sep_forall_r {A} (Φ : A  PROP) Q : ( a, Φ a)  Q   a, Φ a  Q.
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.

Global Instance wand_iff_ne : NonExpansive2 (@bi_wand_iff PROP).
Proof. solve_proper. Qed.
Global Instance wand_iff_proper :
  Proper (() ==> () ==> ()) (@bi_wand_iff PROP) := ne_proper_2 _.

Lemma wand_iff_refl P : emp  P - P.
Proof. apply and_intro; apply wand_intro_l; by rewrite right_id. Qed.

Lemma wand_entails P Q : (P - Q)%I  P  Q.
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Proof. intros. rewrite -[P]emp_sep. by apply wand_elim_l'. Qed.
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Lemma entails_wand P Q : (P  Q)  (P - Q)%I.
Proof. intros ->. apply wand_intro_r. by rewrite left_id. Qed.
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(* A version that works with rewrite, in which bi_emp_valid is unfolded. *)
Lemma entails_wand' P Q : (P  Q)  emp  (P - Q).
Proof. apply entails_wand. Qed.
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Lemma equiv_wand_iff P Q : (P  Q)  (P - Q)%I.
Proof. intros ->; apply wand_iff_refl. Qed.
Lemma wand_iff_equiv P Q : (P - Q)%I  (P  Q).
Proof.
  intros HPQ; apply (anti_symm ());
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    apply wand_entails; rewrite /bi_emp_valid HPQ /bi_wand_iff; auto.
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Qed.

Lemma entails_impl P Q : (P  Q)  (P  Q)%I.
Proof. intros ->. apply impl_intro_l. auto. Qed.
Lemma impl_entails P Q `{!Affine P} : (P  Q)%I  P  Q.
Proof. intros HPQ. apply impl_elim with P=>//. by rewrite {1}(affine P). Qed.

Lemma equiv_iff P Q : (P  Q)  (P  Q)%I.
Proof. intros ->; apply iff_refl. Qed.
Lemma iff_equiv P Q `{!Affine P, !Affine Q} : (P  Q)%I  (P  Q).
Proof.
  intros HPQ; apply (anti_symm ());
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    apply: impl_entails; rewrite /bi_emp_valid HPQ /bi_iff; auto.
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Qed.

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Lemma and_parallel P1 P2 Q1 Q2 :
  (P1  P2) - ((P1 - Q1)  (P2 - Q2)) - Q1  Q2.
Proof.
  apply wand_intro_r, and_intro.
  - rewrite !and_elim_l wand_elim_r. done.
  - rewrite !and_elim_r wand_elim_r. done.
Qed.

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Lemma wandM_sound (mP : option PROP) Q :
  (mP -? Q)  (default emp mP - Q).
Proof. destruct mP; simpl; first done. rewrite emp_wand //. Qed.

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(* Pure stuff *)
Lemma pure_elim φ Q R : (Q  ⌜φ⌝)  (φ  Q  R)  Q  R.
Proof.
  intros HQ HQR. rewrite -(idemp ()%I Q) {1}HQ.
  apply impl_elim_l', pure_elim'=> ?. apply impl_intro_l.
  rewrite and_elim_l; auto.
Qed.
Lemma pure_mono φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
Proof. auto using pure_elim', pure_intro. Qed.
Global Instance pure_mono' : Proper (impl ==> ()) (@bi_pure PROP).
Proof. intros φ1 φ2; apply pure_mono. Qed.
Global Instance pure_flip_mono : Proper (flip impl ==> flip ()) (@bi_pure PROP).
Proof. intros φ1 φ2; apply pure_mono. Qed.
Lemma pure_iff φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
Proof. intros [??]; apply (anti_symm _); auto using pure_mono. Qed.
Lemma pure_elim_l φ Q R : (φ  Q  R)  ⌜φ⌝  Q  R.
Proof. intros; apply pure_elim with φ; eauto. Qed.
Lemma pure_elim_r φ Q R : (φ  Q  R)  Q  ⌜φ⌝  R.
Proof. intros; apply pure_elim with φ; eauto. Qed.

Lemma pure_True (φ : Prop) : φ  ⌜φ⌝  True.
Proof. intros; apply (anti_symm _); auto. Qed.
Lemma pure_False (φ : Prop) : ¬φ  ⌜φ⌝  False.
Proof. intros; apply (anti_symm _); eauto using pure_mono. Qed.

Lemma pure_and φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
Proof.
  apply (anti_symm _).
  - apply and_intro; apply pure_mono; tauto.
  - eapply (pure_elim φ1); [auto|]=> ?. rewrite and_elim_r. auto using pure_mono.
Qed.
Lemma pure_or φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[?|?]; auto using pure_mono.
  - apply or_elim; eauto using pure_mono.
Qed.
Lemma pure_impl φ1 φ2 : ⌜φ1  φ2  (⌜φ1  ⌜φ2).
Proof.
  apply (anti_symm _).
  - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver.
  - rewrite -pure_forall_2. apply forall_intro=> ?.
    by rewrite -(left_id True bi_and (_→_))%I (pure_True φ1) // impl_elim_r.
Qed.
Lemma pure_forall {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
Proof.
  apply (anti_symm _); auto using pure_forall_2.
  apply forall_intro=> x. eauto using pure_mono.
Qed.
Lemma pure_exist {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[x ?]. rewrite -(exist_intro x); auto using pure_mono.
  - apply exist_elim=> x. eauto using pure_mono.
Qed.

Lemma pure_impl_forall φ P : (⌜φ⌝  P)  ( _ : φ, P).
Proof.
  apply (anti_symm _).
  - apply forall_intro=> ?. by rewrite pure_True // left_id.
  - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ).
Qed.
Lemma pure_alt φ : ⌜φ⌝   _ : φ, True.
Proof.
  apply (anti_symm _).
  - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto.
  - by apply exist_elim, pure_intro.
Qed.
Lemma pure_wand_forall φ P `{!Absorbing P} : (⌜φ⌝ - P)  ( _ : φ, P).
Proof.
  apply (anti_symm _).
  - apply forall_intro=> Hφ.
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    rewrite -(pure_intro φ emp%I) // emp_wand //.
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  - apply wand_intro_l, wand_elim_l', pure_elim'=> Hφ.
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    apply wand_intro_l. rewrite (forall_elim Hφ) comm. by apply absorbing.
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Qed.

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(* Properties of the affinely modality *)
Global Instance affinely_ne : NonExpansive (@bi_affinely PROP).
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Proof. solve_proper. Qed.
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Global Instance affinely_proper : Proper (() ==> ()) (@bi_affinely PROP).
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Proof. solve_proper. Qed.
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Global Instance affinely_mono' : Proper (() ==> ()) (@bi_affinely PROP).
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Proof. solve_proper. Qed.
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Global Instance affinely_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_affinely PROP).
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Proof. solve_proper. Qed.

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Lemma affinely_elim_emp P : <affine> P  emp.
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Proof. rewrite /bi_affinely; auto. Qed.
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Lemma affinely_elim P : <affine> P  P.
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Proof. rewrite /bi_affinely; auto. Qed.
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Lemma affinely_mono P Q : (P  Q)  <affine> P  <affine> Q.
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Proof. by intros ->. Qed.
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Lemma affinely_idemp P : <affine> <affine> P  <affine> P.
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Proof. by rewrite /bi_affinely assoc idemp. Qed.
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Lemma affinely_intro' P Q : (<affine> P  Q)  <affine> P  <affine> Q.
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Proof. intros <-. by rewrite affinely_idemp. Qed.
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Lemma affinely_False : <affine> False  False.
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Proof. by rewrite /bi_affinely right_absorb. Qed.
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Lemma affinely_emp : <affine> emp  emp.
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Proof. by rewrite /bi_affinely (idemp bi_and). Qed.
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Lemma affinely_or P Q : <affine> (P  Q)  <affine> P  <affine> Q.
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Proof. by rewrite /bi_affinely and_or_l. Qed.
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Lemma affinely_and P Q : <affine> (P  Q)  <affine> P  <affine> Q.
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Proof.
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  rewrite /bi_affinely -(comm _ P) (assoc _ (_  _)%I) -!(assoc _ P).
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  by rewrite idemp !assoc (comm _ P).
Qed.
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Lemma affinely_sep_2 P Q : <affine> P  <affine> Q  <affine> (P  Q).
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Proof.
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  rewrite /bi_affinely. apply and_intro.
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  - by rewrite !and_elim_l right_id.
  - by rewrite !and_elim_r.
Qed.
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Lemma affinely_sep `{BiPositive PROP} P Q :
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  <affine> (P  Q)  <affine> P  <affine> Q.
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Proof.
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  apply (anti_symm _), affinely_sep_2.
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  by rewrite -{1}affinely_idemp bi_positive !(comm _ (<affine> P)%I) bi_positive.
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Qed.
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Lemma affinely_forall {A} (Φ : A  PROP) : <affine> ( a, Φ a)   a, <affine> (Φ a).
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Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
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Lemma affinely_exist {A} (Φ : A  PROP) : <affine> ( a, Φ a)   a, <affine> (Φ a).
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Proof. by rewrite /bi_affinely and_exist_l. Qed.

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Lemma affinely_True_emp : <affine> True  <affine> emp.
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Proof. apply (anti_symm _); rewrite /bi_affinely; auto. Qed.

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Lemma affinely_and_l P Q : <affine> P  Q  <affine> (P  Q).
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Proof. by rewrite /bi_affinely assoc. Qed.
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Lemma affinely_and_r P Q : P  <affine> Q  <affine> (P  Q).
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Proof. by rewrite /bi_affinely !assoc (comm _ P). Qed.
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Lemma affinely_and_lr P Q : <affine> P  Q  P  <affine> Q.
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Proof. by rewrite affinely_and_l affinely_and_r. Qed.

(* Properties of the absorbingly modality *)
Global Instance absorbingly_ne : NonExpansive (@bi_absorbingly PROP).
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Proof. solve_proper. Qed.
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Global Instance absorbingly_proper : Proper (() ==> ()) (@bi_absorbingly PROP).
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Proof. solve_proper. Qed.
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Global Instance absorbingly_mono' : Proper (() ==> ()) (@bi_absorbingly PROP).
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Proof. solve_proper. Qed.
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Global Instance absorbingly_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_absorbingly PROP).
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Proof. solve_proper. Qed.

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Lemma absorbingly_intro P : P  <absorb> P.
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Proof. by rewrite /bi_absorbingly -True_sep_2. Qed.
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Lemma absorbingly_mono P Q : (P  Q)  <absorb> P  <absorb> Q.
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Proof. by intros ->. Qed.
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Lemma absorbingly_idemp P : <absorb> <absorb> P  <absorb> P.
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Proof.
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  apply (anti_symm _), absorbingly_intro.
  rewrite /bi_absorbingly assoc. apply sep_mono; auto.
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Qed.

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Lemma absorbingly_pure φ : <absorb>  φ    φ .
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Proof.
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  apply (anti_symm _), absorbingly_intro.
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  apply wand_elim_r', pure_elim'=> ?. apply wand_intro_l; auto.
Qed.
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Lemma absorbingly_or P Q : <absorb> (P  Q)  <absorb> P  <absorb> Q.
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Proof. by rewrite /bi_absorbingly sep_or_l. Qed.
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Lemma absorbingly_and_1 P Q : <absorb> (P  Q)  <absorb> P  <absorb> Q.
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Proof. apply and_intro; apply absorbingly_mono; auto. Qed.
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Lemma absorbingly_forall {A} (Φ : A  PROP) : <absorb> ( a, Φ a)   a, <absorb> (Φ a).
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Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
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Lemma absorbingly_exist {A} (Φ : A  PROP) : <absorb> ( a, Φ a)   a, <absorb> (Φ a).
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Proof. by rewrite /bi_absorbingly sep_exist_l. Qed.
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Lemma absorbingly_sep P Q : <absorb> (P  Q)  <absorb> P  <absorb> Q.
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Proof. by rewrite -{1}absorbingly_idemp /bi_absorbingly !assoc -!(comm _ P) !assoc. Qed.
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Lemma absorbingly_True_emp : <absorb> True  <absorb> emp.
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Proof. by rewrite absorbingly_pure /bi_absorbingly right_id. Qed.
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Lemma absorbingly_wand P Q : <absorb> (P - Q)  <absorb> P - <absorb> Q.
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Proof. apply wand_intro_l. by rewrite -absorbingly_sep wand_elim_r. Qed.
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Lemma absorbingly_sep_l P Q : <absorb> P  Q  <absorb> (P  Q).
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Proof. by rewrite /bi_absorbingly assoc. Qed.
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Lemma absorbingly_sep_r P Q : P  <absorb> Q  <absorb> (P  Q).
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Proof. by rewrite /bi_absorbingly !assoc (comm _ P). Qed.
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Lemma absorbingly_sep_lr P Q : <absorb> P  Q  P  <absorb> Q.
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Proof. by rewrite absorbingly_sep_l absorbingly_sep_r. Qed.
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Lemma affinely_absorbingly_elim `{!BiPositive PROP} P : <affine> <absorb> P  <affine> P.
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Proof.
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  apply (anti_symm _), affinely_mono, absorbingly_intro.
  by rewrite /bi_absorbingly affinely_sep affinely_True_emp affinely_emp left_id.
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Qed.

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(* Affine and absorbing propositions *)
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Global Instance Affine_proper : Proper (() ==> iff) (@Affine PROP).
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Proof. solve_proper. Qed.
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Global Instance Absorbing_proper : Proper (() ==> iff) (@Absorbing PROP).
Proof. solve_proper. Qed.
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Lemma affine_affinely P `{!Affine P} : <affine> P  P.
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Proof. rewrite /bi_affinely. apply (anti_symm _); auto. Qed.
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Lemma absorbing_absorbingly P `{!Absorbing P} : <absorb> P  P.
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Proof. by apply (anti_symm _), absorbingly_intro. Qed.
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Lemma True_affine_all_affine P : Affine (PROP:=PROP) True  Affine P.
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Proof. rewrite /Affine=> <-; auto. Qed.
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Lemma emp_absorbing_all_absorbing P : Absorbing (PROP:=PROP) emp  Absorbing P.
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Proof.
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  intros. rewrite /Absorbing -{2}(emp_sep P).
  rewrite -(absorbing emp) absorbingly_sep_l left_id //.
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Qed.
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Lemma sep_elim_l P Q `{H : TCOr (Affine Q) (Absorbing P)} : P  Q  P.
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Proof.
  destruct H.
  - by rewrite (affine Q) right_id.
  - by rewrite (True_intro Q) comm.
Qed.
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Lemma sep_elim_r P Q `{H : TCOr (Affine P) (Absorbing Q)} : P  Q  Q.
Proof. by rewrite comm sep_elim_l. Qed.

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Lemma sep_and P Q :
  TCOr (Affine P) (Absorbing Q)  TCOr (Absorbing P) (Affine Q) 
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  P  Q  P  Q.
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Proof.
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  intros [?|?] [?|?];
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    apply and_intro; apply: sep_elim_l || apply: sep_elim_r.
Qed.
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Lemma affinely_intro P Q `{!Affine P} : (P  Q)  P  <affine> Q.
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Proof. intros <-. by rewrite affine_affinely. Qed.
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Lemma emp_and P `{!Affine P} : emp  P  P.
Proof. apply (anti_symm _); auto. Qed.
Lemma and_emp P `{!Affine P} : P  emp  P.
Proof. apply (anti_symm _); auto. Qed.
Lemma emp_or P `{!Affine P} : emp  P  emp.
Proof. apply (anti_symm _); auto. Qed.
Lemma or_emp P `{!Affine P} : P  emp  emp.
Proof. apply (anti_symm _); auto. Qed.

Lemma True_sep P `{!Absorbing P} : True  P  P.
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Proof. apply (anti_symm _); auto using True_sep_2. Qed.
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Lemma sep_True P `{!Absorbing P} : P  True  P.
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Proof. by rewrite comm True_sep. Qed.
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Lemma True_emp_iff_BiAffine :
  BiAffine PROP  (True  emp).
Proof.
  split.
  - intros ?. exact: affine.
  - rewrite /BiAffine /Affine=>Hemp ?. rewrite -Hemp.
    exact: True_intro.
Qed.

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Section bi_affine.
  Context `{BiAffine PROP}.
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  Global Instance bi_affine_absorbing P : Absorbing P | 0.
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  Proof. by rewrite /Absorbing /bi_absorbingly (affine True%I) left_id. Qed.
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  Global Instance bi_affine_positive : BiPositive PROP.
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  Proof. intros P Q. by rewrite !affine_affinely. Qed.
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  Lemma True_emp : True  emp.
  Proof. apply (anti_symm _); auto using affine. Qed.

  Global Instance emp_and' : LeftId () emp%I (@bi_and PROP).
  Proof. intros P. by rewrite -True_emp left_id. Qed.
  Global Instance and_emp' : RightId () emp%I (@bi_and PROP).
  Proof. intros P. by rewrite -True_emp right_id. Qed.

  Global Instance True_sep' : LeftId () True%I (@bi_sep PROP).
  Proof. intros P. by rewrite True_emp left_id. Qed.
  Global Instance sep_True' : RightId () True%I (@bi_sep PROP).
  Proof. intros P. by rewrite True_emp right_id. Qed.

  Lemma impl_wand_1 P Q : (P  Q)  P - Q.
  Proof. apply wand_intro_l. by rewrite sep_and impl_elim_r. Qed.

  Lemma decide_emp φ `{!Decision φ} (P : PROP) :
    (if decide φ then P else emp)  (⌜φ⌝  P).
  Proof.
    destruct (decide _).
    - by rewrite pure_True // True_impl.
    - by rewrite pure_False // False_impl True_emp.
  Qed.
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End bi_affine.
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(* Properties of the persistence modality *)
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Hint Resolve persistently_mono.
Global Instance persistently_mono' : Proper (() ==> ()) (@bi_persistently PROP).
Proof. intros P Q; apply persistently_mono. Qed.
Global Instance persistently_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_persistently PROP).
Proof. intros P Q; apply persistently_mono. Qed.
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Lemma absorbingly_elim_persistently P : <absorb> <pers> P  <pers> P.
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Proof.
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  apply (anti_symm _), absorbingly_intro.
  by rewrite /bi_absorbingly comm persistently_absorbing.
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Qed.
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Lemma persistently_forall {A} (Ψ : A  PROP) :
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  <pers> ( a, Ψ a)   a, <pers> (Ψ a).
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Proof.
  apply (anti_symm _); auto using persistently_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
Lemma persistently_exist {A} (Ψ : A  PROP) :
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  <pers> ( a, Ψ a)   a, <pers> (Ψ a).
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Proof.
  apply (anti_symm _); auto using persistently_exist_1.
  apply exist_elim=> x. by rewrite (exist_intro x).
Qed.
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Lemma persistently_and P Q : <pers> (P  Q)  <pers> P  <pers> Q.
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Proof. rewrite !and_alt persistently_forall. by apply forall_proper=> -[]. Qed.
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Lemma persistently_or P Q : <pers> (P  Q)  <pers> P  <pers> Q.
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Proof. rewrite !or_alt persistently_exist. by apply exist_proper=> -[]. Qed.
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Lemma persistently_impl P Q : <pers> (P  Q)  <pers> P  <pers> Q.
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Proof.
  apply impl_intro_l; rewrite -persistently_and.
  apply persistently_mono, impl_elim with P; auto.
Qed.

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Lemma persistently_emp_intro P : P  <pers> emp.
Proof.
  by rewrite -(left_id emp%I bi_sep P) {1}persistently_emp_2 persistently_absorbing.
Qed.

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Lemma persistently_True_emp : <pers> True  <pers> emp.
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Proof. apply (anti_symm _); auto using persistently_emp_intro. Qed.

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Lemma persistently_and_emp P : <pers> P  <pers> (emp  P).
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Proof.
  apply (anti_symm ()); last by rewrite and_elim_r.
  rewrite persistently_and. apply and_intro; last done.
  apply persistently_emp_intro.
Qed.

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Lemma persistently_and_sep_elim_emp P Q : <pers> P  Q  (emp  P)  Q.
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Proof.
  rewrite persistently_and_emp.
  apply persistently_and_sep_elim.
Qed.

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Lemma persistently_and_sep_assoc P Q R : <pers> P  (Q  R)  (<pers> P  Q)  R.
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Proof.
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  apply (anti_symm ()).
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  - rewrite {1}persistently_idemp_2 persistently_and_sep_elim_emp assoc.
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    apply sep_mono_l, and_intro.
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    + by rewrite and_elim_r persistently_absorbing.
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    + by rewrite and_elim_l left_id.
  - apply and_intro.
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    + by rewrite and_elim_l persistently_absorbing.
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    + by rewrite and_elim_r.
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Qed.
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Lemma persistently_and_emp_elim P : emp  <pers> P  P.
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Proof. by rewrite comm persistently_and_sep_elim_emp right_id and_elim_r. Qed.
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Lemma persistently_into_absorbingly P : <pers> P  <absorb> P.
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Proof.
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  rewrite -(right_id True%I _ (<pers> _)%I) -{1}(emp_sep True%I).
  rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim comm //.
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Qed.
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Lemma persistently_elim P `{!Absorbing P} : <pers> P  P.
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Proof. by rewrite persistently_into_absorbingly absorbing_absorbingly. Qed.
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Lemma persistently_idemp_1 P : <pers> <pers> P  <pers> P.
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Proof. by rewrite persistently_into_absorbingly absorbingly_elim_persistently. Qed.
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Lemma persistently_idemp P : <pers> <pers> P  <pers> P.
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Proof. apply (anti_symm _); auto using persistently_idemp_1, persistently_idemp_2. Qed.
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Lemma persistently_intro' P Q : (<pers> P  Q)  <pers> P  <pers> Q.
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Proof. intros <-. apply persistently_idemp_2. Qed.

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Lemma persistently_pure φ : <pers> ⌜φ⌝  ⌜φ⌝.
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Proof.
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  apply (anti_symm _).
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  { by rewrite persistently_into_absorbingly absorbingly_pure. }
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  apply pure_elim'=> Hφ.
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  trans ( x : False, <pers> True : PROP)%I; [by apply forall_intro|].
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  rewrite persistently_forall_2. auto using persistently_mono, pure_intro.
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Qed.

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Lemma persistently_sep_dup P : <pers> P  <pers> P  <pers> P.
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Proof.
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  apply (anti_symm _).
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  - rewrite -{1}(idemp bi_and (<pers> _)%I).
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    by rewrite -{2}(emp_sep (<pers> _)%I)
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      persistently_and_sep_assoc and_elim_l.
  - by rewrite persistently_absorbing.
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Qed.

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Lemma persistently_and_sep_l_1 P Q : <pers> P  Q  <pers> P  Q.
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Proof.
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  by rewrite -{1}(emp_sep Q%I) persistently_and_sep_assoc and_elim_l.
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Qed.
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Lemma persistently_and_sep_r_1 P Q : P  <pers> Q  P  <pers> Q.
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Proof. by rewrite !(comm _ P) persistently_and_sep_l_1. Qed.

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Lemma persistently_and_sep P Q : <pers> (P  Q)  <pers> (P  Q).
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Proof.
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  rewrite persistently_and.
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  rewrite -{1}persistently_idemp -persistently_and -{1}(emp_sep Q%I).
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  by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim.
Qed.

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Lemma persistently_affinely_elim P : <pers> <affine> P  <pers> P.
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Proof.
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  by rewrite /bi_affinely persistently_and -persistently_True_emp
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             persistently_pure left_id.
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Qed.

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Lemma and_sep_persistently P Q : <pers> P  <pers> Q  <pers> P  <pers> Q.
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Proof.
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  apply (anti_symm _); auto using persistently_and_sep_l_1.
  apply and_intro.
  - by rewrite persistently_absorbing.
  - by rewrite comm persistently_absorbing.
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Qed.
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Lemma persistently_sep_2 P Q : <pers> P  <pers> Q  <pers> (P  Q).
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Proof. by rewrite -persistently_and_sep persistently_and -and_sep_persistently. Qed.
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Lemma persistently_sep `{BiPositive PROP} P Q : <pers> (P  Q)  <pers> P  <pers> Q.
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Proof.
  apply (anti_symm _); auto using persistently_sep_2.
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  rewrite -persistently_affinely_elim affinely_sep -and_sep_persistently. apply and_intro.
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  - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
  - by rewrite (affinely_elim_emp P) left_id affinely_elim.
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