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From iris.bi Require Export interface.
From iris.algebra Require Import monoid.
From stdpp Require Import hlist.

Definition bi_iff {PROP : bi} (P Q : PROP) : PROP := ((P  Q)  (Q  P))%I.
Arguments bi_iff {_} _%I _%I : simpl never.
Instance: Params (@bi_iff) 1.
Infix "↔" := bi_iff : bi_scope.

Definition bi_wand_iff {PROP : bi} (P Q : PROP) : PROP :=
  ((P - Q)  (Q - P))%I.
Arguments bi_wand_iff {_} _%I _%I : simpl never.
Instance: Params (@bi_wand_iff) 1.
Infix "∗-∗" := bi_wand_iff (at level 95, no associativity) : bi_scope.

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Class Persistent {PROP : bi} (P : PROP) := persistent : P   P.
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Arguments Persistent {_} _%I : simpl never.
Arguments persistent {_} _%I {_}.
Hint Mode Persistent + ! : typeclass_instances.
Instance: Params (@Persistent) 1.

Definition bi_bare {PROP : bi} (P : PROP) : PROP := (emp  P)%I.
Arguments bi_bare {_} _%I : simpl never.
Instance: Params (@bi_bare) 1.
Typeclasses Opaque bi_bare.
Notation "■ P" := (bi_bare P) (at level 20, right associativity) : bi_scope.
Notation "⬕ P" := (  P)%I (at level 20, right associativity) : bi_scope.

Class Affine {PROP : bi} (Q : PROP) := affine : Q  emp.
Arguments Affine {_} _%I : simpl never.
Arguments affine {_} _%I {_}.
Hint Mode Affine + ! : typeclass_instances.

Class AffineBI (PROP : bi) := absorbing_bi (Q : PROP) : Affine Q.
Existing Instance absorbing_bi | 0.

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Class PositiveBI (PROP : bi) :=
  positive_bi (P Q : PROP) :  (P  Q)   P  Q.

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Definition bi_sink {PROP : bi} (P : PROP) : PROP := (True  P)%I.
Arguments bi_sink {_} _%I : simpl never.
Instance: Params (@bi_sink) 1.
Typeclasses Opaque bi_sink.
Notation "▲ P" := (bi_sink P) (at level 20, right associativity) : bi_scope.

Class Absorbing {PROP : bi} (P : PROP) := absorbing :  P  P.
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Arguments Absorbing {_} _%I : simpl never.
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Arguments absorbing {_} _%I.
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Definition bi_persistently_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
  (if p then  P else P)%I.
Arguments bi_persistently_if {_} !_ _%I /.
Instance: Params (@bi_persistently_if) 2.
Typeclasses Opaque bi_persistently_if.
Notation "□? p P" := (bi_persistently_if p P)
  (at level 20, p at level 9, P at level 20,
   right associativity, format "□? p  P") : bi_scope.

Definition bi_bare_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
  (if p then  P else P)%I.
Arguments bi_bare_if {_} !_ _%I /.
Instance: Params (@bi_bare_if) 2.
Typeclasses Opaque bi_bare_if.
Notation "■? p P" := (bi_bare_if p P)
  (at level 20, p at level 9, P at level 20,
   right associativity, format "■? p  P") : bi_scope.
Notation "⬕? p P" := (?p ?p P)%I
  (at level 20, p at level 9, P at level 20,
   right associativity, format "⬕? p  P") : bi_scope.

Fixpoint bi_hexist {PROP : bi} {As} : himpl As PROP  PROP :=
  match As return himpl As PROP  PROP with
  | tnil => id
  | tcons A As => λ Φ,  x, bi_hexist (Φ x)
  end%I.
Fixpoint bi_hforall {PROP : bi} {As} : himpl As PROP  PROP :=
  match As return himpl As PROP  PROP with
  | tnil => id
  | tcons A As => λ Φ,  x, bi_hforall (Φ x)
  end%I.

Definition bi_laterN {PROP : sbi} (n : nat) (P : PROP) : PROP :=
  Nat.iter n bi_later P.
Arguments bi_laterN {_} !_%nat_scope _%I.
Instance: Params (@bi_laterN) 2.
Notation "▷^ n P" := (bi_laterN n P)
  (at level 20, n at level 9, P at level 20, format "▷^ n  P") : bi_scope.
Notation "▷? p P" := (bi_laterN (Nat.b2n p) P)
  (at level 20, p at level 9, P at level 20, format "▷? p  P") : bi_scope.

Definition bi_except_0 {PROP : sbi} (P : PROP) : PROP := ( False  P)%I.
Arguments bi_except_0 {_} _%I : simpl never.
Notation "◇ P" := (bi_except_0 P) (at level 20, right associativity) : bi_scope.
Instance: Params (@bi_except_0) 1.
Typeclasses Opaque bi_except_0.

Class Timeless {PROP : sbi} (P : PROP) := timeless :  P   P.
Arguments Timeless {_} _%I : simpl never.
Arguments timeless {_} _%I {_}.
Hint Mode Timeless + ! : typeclass_instances.
Instance: Params (@Timeless) 1.

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 *)
Notation "P ⊢ Q" := (@bi_entails PROP P%I Q%I).
Notation "P ⊣⊢ Q" := (equiv (A:=bi_car PROP) P%I Q%I).

(* 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.
 Global Instance bi_valid_proper : Proper (() ==> iff) (@bi_valid PROP).
Proof. solve_proper. Qed.
Global Instance bi_valid_mono : Proper (() ==> impl) (@bi_valid PROP).
Proof. solve_proper. Qed.
Global Instance bi_valid_flip_mono :
  Proper (flip () ==> flip impl) (@bi_valid PROP).
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 internal_eq_proper (A : ofeT) :
  Proper (() ==> () ==> ()) (@bi_internal_eq PROP A) := ne_proper_2 _.
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.

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.

Lemma exists_impl_forall {A} P (Ψ : A  PROP) :
  (( 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.

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.

(* Equality stuff *)
Hint Resolve internal_eq_refl.
Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a  b  P  a  b.
Proof. intros ->. auto. Qed.
Lemma internal_eq_rewrite' {A : ofeT} a b (Ψ : A  PROP) P
  {HΨ : NonExpansive Ψ} : (P  a  b)  (P  Ψ a)  P  Ψ b.
Proof.
  intros Heq HΨa. rewrite -(idemp bi_and P) {1}Heq HΨa.
  apply impl_elim_l'. by apply internal_eq_rewrite.
Qed.

Lemma internal_eq_sym {A : ofeT} (a b : A) : a  b  b  a.
Proof. apply (internal_eq_rewrite' a b (λ b, b  a)%I); auto. Qed.
Lemma internal_eq_iff P Q : P  Q  P  Q.
Proof. apply (internal_eq_rewrite' P Q (λ Q, P  Q))%I; auto using iff_refl. Qed.

Lemma f_equiv {A B : ofeT} (f : A  B) `{!NonExpansive f} x y :
  x  y  f x  f y.
Proof. apply (internal_eq_rewrite' x y (λ y, f x  f y)%I); auto. Qed.

Lemma prod_equivI {A B : ofeT} (x y : A * B) : x  y  x.1  y.1  x.2  y.2.
Proof.
  apply (anti_symm _).
  - apply and_intro; apply f_equiv; apply _.
  - rewrite {3}(surjective_pairing x) {3}(surjective_pairing y).
    apply (internal_eq_rewrite' (x.1) (y.1) (λ a, (x.1,x.2)  (a,y.2))%I); auto.
    apply (internal_eq_rewrite' (x.2) (y.2) (λ b, (x.1,x.2)  (x.1,b))%I); auto.
Qed.
Lemma sum_equivI {A B : ofeT} (x y : A + B) :
  x  y 
    match x, y with
    | inl a, inl a' => a  a' | inr b, inr b' => b  b' | _, _ => False
    end.
Proof.
  apply (anti_symm _).
  - apply (internal_eq_rewrite' x y (λ y,
             match x, y with
             | inl a, inl a' => a  a' | inr b, inr b' => b  b' | _, _ => False
             end)%I); auto.
    destruct x; auto.
  - destruct x as [a|b], y as [a'|b']; auto; apply f_equiv, _.
Qed.
Lemma option_equivI {A : ofeT} (x y : option A) :
  x  y  match x, y with
           | Some a, Some a' => a  a' | None, None => True | _, _ => False
           end.
Proof.
  apply (anti_symm _).
  - apply (internal_eq_rewrite' x y (λ y,
             match x, y with
             | Some a, Some a' => a  a' | None, None => True | _, _ => False
             end)%I); auto.
    destruct x; auto.
  - destruct x as [a|], y as [a'|]; auto. apply f_equiv, _.
Qed.

Lemma sig_equivI {A : ofeT} (P : A  Prop) (x y : sig P) : `x  `y  x  y.
Proof. apply (anti_symm _). apply sig_eq. apply f_equiv, _. Qed.

Lemma ofe_funC_equivI {A B} (f g : A -c> B) : f  g   x, f x  g x.
Proof.
  apply (anti_symm _); auto using fun_ext.
  apply (internal_eq_rewrite' f g (λ g,  x : A, f x  g x)%I); auto.
  intros n h h' Hh; apply forall_ne=> x; apply internal_eq_ne; auto.
Qed.
Lemma ofe_morC_equivI {A B : ofeT} (f g : A -n> B) : f  g   x, f x  g x.
Proof.
  apply (anti_symm _).
  - apply (internal_eq_rewrite' f g (λ g,  x : A, f x  g x)%I); auto.
  - rewrite -(ofe_funC_equivI (ofe_mor_car _ _ f) (ofe_mor_car _ _ g)).
    set (h1 (f : A -n> B) :=
      exist (λ f : A -c> B, NonExpansive f) f (ofe_mor_ne A B f)).
    set (h2 (f : sigC (λ f : A -c> B, NonExpansive f)) :=
      @CofeMor A B (`f) (proj2_sig f)).
    assert ( f, h2 (h1 f) = f) as Hh by (by intros []).
    assert (NonExpansive h2) by (intros ??? EQ; apply EQ).
    by rewrite -{2}[f]Hh -{2}[g]Hh -f_equiv -sig_equivI.
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.

Lemma sep_intro_valid_l P Q R : P  (R  Q)  R  P  Q.
Proof. intros ? ->. rewrite -{1}(left_id emp%I _ Q). by apply sep_mono. Qed.
Lemma sep_intro_valid_r P Q R : (R  P)  Q  R  P  Q.
Proof. intros -> ?. rewrite comm. by apply sep_intro_valid_l. Qed.
Lemma sep_elim_valid_l P Q R : P  (P  R  Q)  R  Q.
Proof. intros <- <-. by rewrite left_id. Qed.
Lemma sep_elim_valid_r P Q R : P  (R  P  Q)  R  Q.
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.

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

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.
Proof. intros. rewrite -[P]left_id. by apply wand_elim_l'. Qed.
Lemma entails_wand P Q : (P  Q)  (P - Q)%I.
Proof. intros ->. apply wand_intro_r. by rewrite left_id. Qed.

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 ());
    apply wand_entails; rewrite /bi_valid HPQ /bi_wand_iff; auto.
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 ());
    apply: impl_entails; rewrite /bi_valid HPQ /bi_iff; auto.
Qed.

(* 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.
    by rewrite -(left_id emp%I _ (_ - _)%I) (pure_intro emp%I φ) // wand_elim_r.
  - 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.

Lemma pure_internal_eq {A : ofeT} (x y : A) : x  y  x  y.
Proof. apply pure_elim'=> ->. apply internal_eq_refl. Qed.
Lemma discrete_eq {A : ofeT} (a b : A) : Discrete a  a  b  a  b.
Proof.
  intros. apply (anti_symm _); auto using discrete_eq_1, pure_internal_eq.
Qed.

(* Properties of the bare modality *)
Global Instance bare_ne : NonExpansive (@bi_bare PROP).
Proof. solve_proper. Qed.
Global Instance bare_proper : Proper (() ==> ()) (@bi_bare PROP).
Proof. solve_proper. Qed.
Global Instance bare_mono' : Proper (() ==> ()) (@bi_bare PROP).
Proof. solve_proper. Qed.
Global Instance bare_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_bare PROP).
Proof. solve_proper. Qed.

Lemma bare_elim_emp P :  P  emp.
Proof. rewrite /bi_bare; auto. Qed.
Lemma bare_elim P :  P  P.
Proof. rewrite /bi_bare; auto. Qed.
Lemma bare_mono P Q : (P  Q)   P   Q.
Proof. by intros ->. Qed.
Lemma bare_idemp P :   P   P.
Proof. by rewrite /bi_bare assoc idemp. Qed.

Lemma bare_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. by rewrite bare_idemp. Qed.

Lemma bare_False :  False  False.
Proof. by rewrite /bi_bare right_absorb. Qed.
Lemma bare_emp :  emp  emp.
Proof. by rewrite /bi_bare (idemp bi_and). Qed.
Lemma bare_or P Q :  (P  Q)   P   Q.
Proof. by rewrite /bi_bare and_or_l. Qed.
Lemma bare_and P Q :  (P  Q)   P   Q.
Proof.
  rewrite /bi_bare -(comm _ P) (assoc _ (_  _)%I) -!(assoc _ P).
  by rewrite idemp !assoc (comm _ P).
Qed.
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Lemma bare_sep_2 P Q :  P   Q   (P  Q).
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Proof.
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  rewrite /bi_bare. apply and_intro.
  - by rewrite !and_elim_l right_id.
  - by rewrite !and_elim_r.
Qed.
Lemma bare_sep `{PositiveBI PROP} P Q :  (P  Q)   P   Q.
Proof.
  apply (anti_symm _), bare_sep_2.
  by rewrite -{1}bare_idemp positive_bi !(comm _ ( P)%I) positive_bi.
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Qed.
Lemma bare_forall {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
Lemma bare_exist {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. by rewrite /bi_bare and_exist_l. Qed.

Lemma bare_True_emp :  True   emp.
Proof. apply (anti_symm _); rewrite /bi_bare; auto. Qed.

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Lemma bare_and_l P Q :  P  Q   (P  Q).
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Proof. by rewrite /bi_bare assoc. Qed.
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Lemma bare_and_r P Q : P   Q   (P  Q).
Proof. by rewrite /bi_bare !assoc (comm _ P). Qed.
Lemma bare_and_lr P Q :  P  Q  P   Q.
Proof. by rewrite bare_and_l bare_and_r. Qed.
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(* Properties of the sink modality *)
Global Instance sink_ne : NonExpansive (@bi_sink PROP).
Proof. solve_proper. Qed.
Global Instance sink_proper : Proper (() ==> ()) (@bi_sink PROP).
Proof. solve_proper. Qed.
Global Instance sink_mono' : Proper (() ==> ()) (@bi_sink PROP).
Proof. solve_proper. Qed.
Global Instance sink_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_sink PROP).
Proof. solve_proper. Qed.

Lemma sink_intro P : P   P.
Proof. by rewrite /bi_sink -True_sep_2. Qed.
Lemma sink_mono P Q : (P  Q)   P   Q.
Proof. by intros ->. Qed.
Lemma sink_idemp P :   P   P.
Proof.
  apply (anti_symm _), sink_intro.
  rewrite /bi_sink assoc. apply sep_mono; auto.
Qed.

Lemma sink_pure φ :   φ    φ .
Proof.
  apply (anti_symm _), sink_intro.
  apply wand_elim_r', pure_elim'=> ?. apply wand_intro_l; auto.
Qed.
Lemma sink_or P Q :  (P  Q)   P   Q.
Proof. by rewrite /bi_sink sep_or_l. Qed.
Lemma sink_and P Q :  (P  Q)   P   Q.
Proof. apply and_intro; apply sink_mono; auto. Qed.
Lemma sink_forall {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
Lemma sink_exist {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. by rewrite /bi_sink sep_exist_l. Qed.

Lemma sink_internal_eq {A : ofeT} (x y : A) :  (x  y)  x  y.
Proof.
  apply (anti_symm _), sink_intro.
  apply wand_elim_r', (internal_eq_rewrite' x y (λ y, True - x  y)%I); auto.
  apply wand_intro_l, internal_eq_refl.
Qed.

Lemma sink_sep P Q :  (P  Q)   P   Q.
Proof. by rewrite -{1}sink_idemp /bi_sink !assoc -!(comm _ P) !assoc. Qed.
Lemma sink_True_emp :  True   emp.
Proof. by rewrite sink_pure /bi_sink right_id. Qed.
Lemma sink_wand P Q :  (P - Q)   P -  Q.
Proof. apply wand_intro_l. by rewrite -sink_sep wand_elim_r. Qed.

Lemma sink_sep_l P Q :  P  Q   (P  Q).
Proof. by rewrite /bi_sink assoc. Qed.
Lemma sink_sep_r P Q : P   Q   (P  Q).
Proof. by rewrite /bi_sink !assoc (comm _ P). Qed.
Lemma sink_sep_lr P Q :  P  Q  P   Q.
Proof. by rewrite sink_sep_l sink_sep_r. Qed.

Lemma bare_sink `{!PositiveBI PROP} P :   P   P.
Proof.
  apply (anti_symm _), bare_mono, sink_intro.
  by rewrite /bi_sink bare_sep bare_True_emp bare_emp left_id.
Qed.

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(* Affine propositions *)
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Global Instance Affine_proper : Proper (() ==> iff) (@Affine PROP).
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Proof. solve_proper. Qed.

Global Instance emp_affine_l : Affine (emp%I : PROP).
Proof. by rewrite /Affine. Qed.
Global Instance and_affine_l P Q : Affine P  Affine (P  Q).
Proof. rewrite /Affine=> ->; auto. Qed.
Global Instance and_affine_r P Q : Affine Q  Affine (P  Q).
Proof. rewrite /Affine=> ->; auto. Qed.
Global Instance or_affine P Q : Affine P  Affine Q  Affine (P  Q).
Proof.  rewrite /Affine=> -> ->; auto. Qed.
Global Instance forall_affine `{Inhabited A} (Φ : A  PROP) :
  ( x, Affine (Φ x))  Affine ( x, Φ x).
Proof. intros. rewrite /Affine (forall_elim inhabitant). apply: affine. Qed.
Global Instance exist_affine {A} (Φ : A  PROP) :
  ( x, Affine (Φ x))  Affine ( x, Φ x).
Proof. rewrite /Affine=> H. apply exist_elim=> a. by rewrite H. Qed.
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Global Instance sep_affine P Q : Affine P  Affine Q  Affine (P  Q).
Proof. rewrite /Affine=>-> ->. by rewrite left_id. Qed.
Global Instance bare_affine P : Affine ( P).
Proof. rewrite /bi_bare. apply _. Qed.

(* Absorbing propositions *)
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Global Instance Absorbing_proper : Proper (() ==> iff) (@Absorbing PROP).
Proof. solve_proper. Qed.
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Global Instance pure_absorbing φ : Absorbing (⌜φ⌝%I : PROP).
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Proof. by rewrite /Absorbing sink_pure. Qed.
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Global Instance and_absorbing P Q : Absorbing P  Absorbing Q  Absorbing (P  Q).
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Proof. intros. by rewrite /Absorbing sink_and !absorbing. Qed.
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Global Instance or_absorbing P Q : Absorbing P  Absorbing Q  Absorbing (P  Q).
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Proof. intros. by rewrite /Absorbing sink_or !absorbing. Qed.
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Global Instance forall_absorbing {A} (Φ : A  PROP) :
  ( x, Absorbing (Φ x))  Absorbing ( x, Φ x).
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Proof. rewrite /Absorbing=> ?. rewrite sink_forall. auto using forall_mono. Qed.
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Global Instance exist_absorbing {A} (Φ : A  PROP) :
  ( x, Absorbing (Φ x))  Absorbing ( x, Φ x).
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Proof. rewrite /Absorbing=> ?. rewrite sink_exist. auto using exist_mono. Qed.
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Global Instance internal_eq_absorbing {A : ofeT} (x y : A) :
  Absorbing (x  y : PROP)%I.
Proof. by rewrite /Absorbing sink_internal_eq. Qed.

Global Instance sep_absorbing_l P Q : Absorbing P  Absorbing (P  Q).
Proof. intros. by rewrite /Absorbing -sink_sep_l absorbing. Qed.
Global Instance sep_absorbing_r P Q : Absorbing Q  Absorbing (P  Q).
Proof. intros. by rewrite /Absorbing -sink_sep_r absorbing. Qed.
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Global Instance wand_absorbing P Q : Absorbing Q  Absorbing (P - Q).
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Proof. intros. by rewrite /Absorbing sink_wand !absorbing -sink_intro. Qed.

Global Instance sink_absorbing P : Absorbing ( P).
Proof. rewrite /bi_sink. apply _. Qed.
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(* Properties of affine and absorbing propositions *)
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Lemma affine_bare P `{!Affine P} :  P  P.
Proof. rewrite /bi_bare. apply (anti_symm _); auto. Qed.
Lemma absorbing_sink P `{!Absorbing P} :  P  P.
Proof. by apply (anti_symm _), sink_intro. Qed.

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Lemma True_affine_all_affine P : Affine (True%I : PROP)  Affine P.
Proof. rewrite /Affine=> <-; auto. Qed.
Lemma emp_absorbing_all_absorbing P : Absorbing (emp%I : PROP)  Absorbing P.
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Proof.
  intros. rewrite /Absorbing -{2}(left_id emp%I _ P).
  by rewrite -(absorbing emp) sink_sep_l left_id.
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
    `{HPQ : TCOr (TCAnd (Affine P) (Affine Q)) (TCAnd (Absorbing P) (Absorbing Q))} :
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  P  Q  P  Q.
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Proof.
  destruct HPQ as [[??]|[??]];
    apply and_intro; apply: sep_elim_l || apply: sep_elim_r.
Qed.
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Lemma bare_intro P Q `{!Affine P} : (P  Q)  P   Q.
Proof. intros <-. by rewrite affine_bare. Qed.

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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Section affine_bi.
  Context `{AffineBI PROP}.

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  Global Instance affine_bi_absorbing P : Absorbing P | 0.
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  Proof. by rewrite /Absorbing /bi_sink (affine True%I) left_id. Qed.
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  Global Instance affine_bi_positive : PositiveBI PROP.
  Proof. intros P Q. by rewrite !affine_bare. 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.
End affine_bi.

(* Properties of the persistently modality *)
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 sink_persistently P :   P   P.
Proof.
  apply (anti_symm _), sink_intro. by rewrite /bi_sink comm persistently_absorbing.
Qed.
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Global Instance persistently_absorbing P : Absorbing ( P).
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Proof. by rewrite /Absorbing sink_persistently. Qed.
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Lemma persistently_and_sep_assoc P Q R :  P  (Q  R)  ( P  Q)  R.
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Proof.
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  apply (anti_symm ()).
  - rewrite {1}persistently_idemp_2 persistently_and_sep_elim assoc.
    apply sep_mono_l, and_intro.
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    + by rewrite and_elim_r sep_elim_l.
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    + by rewrite and_elim_l left_id.
  - apply and_intro.
    + by rewrite and_elim_l sep_elim_l.
    + by rewrite and_elim_r.
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Qed.
Lemma persistently_and_emp_elim P : emp   P  P.
Proof. by rewrite comm persistently_and_sep_elim right_id and_elim_r. Qed.
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Lemma persistently_elim_sink P :  P   P.
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Proof.
  rewrite -(right_id True%I _ ( _)%I) -{1}(left_id emp%I _ True%I).
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  by 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} :  P  P.
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Proof. by rewrite persistently_elim_sink absorbing_sink. Qed.
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Lemma persistently_idemp_1 P :   P   P.
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Proof. by rewrite persistently_elim_sink sink_persistently. Qed.
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Lemma persistently_idemp P :   P   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 : ( P  Q)   P   Q.
Proof. intros <-. apply persistently_idemp_2. Qed.

Lemma persistently_pure φ :  ⌜φ⌝  ⌜φ⌝.
Proof.
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  apply (anti_symm _); first by rewrite persistently_elim.
  apply pure_elim'=> Hφ.
  trans ( x : False,  True : PROP)%I; [by apply forall_intro|].
  rewrite persistently_forall_2. auto using persistently_mono, pure_intro.
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Qed.
Lemma persistently_forall {A} (Ψ : A  PROP) : (  a, Ψ a)  ( a,  Ψ a).
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) : (  a, Ψ a)  ( a,  Ψ a).
Proof.
  apply (anti_symm _); auto using persistently_exist_1.
  apply exist_elim=> x. by rewrite (exist_intro x).
Qed.
Lemma persistently_and P Q :  (P  Q)   P   Q.
Proof. rewrite !and_alt persistently_forall. by apply forall_proper=> -[]. Qed.
Lemma persistently_or P Q :  (P  Q)   P   Q.
Proof. rewrite !or_alt persistently_exist. by apply exist_proper=> -[]. Qed.
Lemma persistently_impl P Q :  (P  Q)   P   Q.
Proof.
  apply impl_intro_l; rewrite -persistently_and.
  apply persistently_mono, impl_elim with P; auto.
Qed.

Lemma persistently_internal_eq {A : ofeT} (a b : A) :  (a  b)  a  b.
Proof.
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  apply (anti_symm ()); first by rewrite persistently_elim.
  apply (internal_eq_rewrite' a b (λ b,  (a  b))%I); auto.
  rewrite -(internal_eq_refl emp%I a). apply persistently_emp_intro.
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Qed.

Lemma persistently_sep_dup P :  P   P   P.
Proof.
  apply (anti_symm _); last by eauto using sep_elim_l with typeclass_instances.
  rewrite -{1}(idemp bi_and ( _)%I) -{2}(left_id emp%I _ ( _)%I).
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  by rewrite persistently_and_sep_assoc and_elim_l.
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Qed.

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

Lemma persistently_True_emp :  True   emp.
Proof. apply (anti_symm _); auto using persistently_emp_intro. Qed.
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Lemma persistently_and_sep P Q :  (P  Q)   (P  Q).
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Proof.
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  rewrite persistently_and.
  rewrite -{1}persistently_idemp -persistently_and -{1}(left_id emp%I _ Q%I).
  by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim.
Qed.

Lemma persistently_bare P :   P   P.
Proof.
  by rewrite /bi_bare persistently_and -persistently_True_emp
             persistently_pure left_id.
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Qed.

Lemma and_sep_persistently P Q :  P   Q   P   Q.
Proof.
  apply (anti_symm _).
  - auto using persistently_and_sep_l_1.
  - eauto 10 using sep_elim_l, sep_elim_r with typeclass_instances.
Qed.
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Lemma persistently_sep_2 P Q :  P   Q   (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 `{PositiveBI PROP} P Q :  (P  Q)   P   Q.
Proof.
  apply (anti_symm _); auto using persistently_sep_2.
  by rewrite -persistently_bare bare_sep sep_and !bare_elim persistently_and
             and_sep_persistently.
Qed.
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Lemma persistently_wand P Q :  (P - Q)   P -  Q.
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Proof. apply wand_intro_r. by rewrite persistently_sep_2 wand_elim_l. Qed.
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Lemma persistently_entails_l P Q : (P   Q)  P   Q  P.
Proof. intros; rewrite -persistently_and_sep_l_1; auto. Qed.
Lemma persistently_entails_r P Q : (P   Q)  P  P   Q.
Proof. intros; rewrite -persistently_and_sep_r_1; auto. Qed.

Lemma persistently_impl_wand_2 P Q :  (P - Q)   (P  Q).
Proof.
  apply persistently_intro', impl_intro_r.
  rewrite -{2}(left_id emp%I _ P%I) persistently_and_sep_assoc.
  by rewrite (comm bi_and) persistently_and_emp_elim wand_elim_l.
Qed.

Section persistently_bare_bi.
  Context `{AffineBI PROP}.

  Lemma persistently_emp :  emp  emp.
  Proof. by rewrite -!True_emp persistently_pure. Qed.

  Lemma persistently_and_sep_l P Q :  P  Q   P  Q.
  Proof.
    apply (anti_symm ());
      eauto using persistently_and_sep_l_1, sep_and with typeclass_instances.
  Qed.
  Lemma persistently_and_sep_r P Q : P   Q  P   Q.
  Proof. by rewrite !(comm _ P) persistently_and_sep_l. Qed.

  Lemma persistently_impl_wand P Q :  (P  Q)   (P - Q).
  Proof.
    apply (anti_symm ()); auto using persistently_impl_wand_2.
    apply persistently_intro', wand_intro_l.
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    by rewrite -persistently_and_sep_r persistently_elim impl_elim_r.
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  Qed.

  Lemma wand_alt P Q : (P - Q)   R, R   (P  R  Q).
  Proof.
    apply (anti_symm ()).
    - rewrite -(right_id True%I bi_sep (P - Q)%I) -(exist_intro (P - Q)%I).
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      apply sep_mono_r. rewrite -persistently_pure.
      apply persistently_intro', impl_intro_l.
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      by rewrite wand_elim_r persistently_pure right_id.
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    - apply exist_elim=> R. apply wand_intro_l.
      rewrite assoc -persistently_and_sep_r.
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      by rewrite persistently_elim impl_elim_r.
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  Qed.
  Lemma impl_alt P Q : (P  Q)   R, R   (P  R - Q).
  Proof.
    apply (anti_symm ()).
    - rewrite -(right_id True%I bi_and (P  Q)%I) -(exist_intro (P  Q)%I).
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      apply and_mono_r. rewrite -persistently_pure.
      apply persistently_intro', wand_intro_l.
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      by rewrite impl_elim_r persistently_pure right_id.
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    - apply exist_elim=> R. apply impl_intro_l. rewrite assoc persistently_and_sep_r.
      by rewrite persistently_elim wand_elim_r.
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  Qed.
End persistently_bare_bi.


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(* The combined bare persistently modality *)
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Lemma bare_persistently_elim P :  P  P.
Proof. apply persistently_and_emp_elim. Qed.
Lemma bare_persistently_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. by rewrite persistently_bare persistently_idemp. Qed.

Lemma bare_persistently_emp :  emp  emp.
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Proof. by rewrite -persistently_True_emp persistently_pure bare_True_emp bare_emp. Qed.
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Lemma bare_persistently_and P Q :  (P  Q)   P   Q.
Proof. by rewrite persistently_and bare_and. Qed.
Lemma bare_persistently_or P Q :  (P  Q)   P   Q.
Proof. by rewrite persistently_or bare_or. Qed.
Lemma bare_persistently_exist {A} (Φ : A  PROP) :  ( x, Φ x)   x,  Φ x.
Proof. by rewrite persistently_exist bare_exist. Qed.
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Lemma bare_persistently_sep_2 P Q :  P   Q   (P  Q).
Proof. by rewrite bare_sep_2 persistently_sep_2. Qed.
Lemma bare_persistently_sep `{PositiveBI PROP} P Q :  (P  Q)   P   Q.
Proof. by rewrite -bare_sep -persistently_sep. Qed.
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Lemma bare_persistently_idemp P :   P   P.
Proof. by rewrite persistently_bare persistently_idemp. Qed.

Lemma persistently_and_bare_sep_l P Q :  P  Q   P  Q.
Proof.
  apply (anti_symm _).
  - by rewrite /bi_bare -(comm bi_and ( P)%I) -persistently_and_sep_assoc left_id.
  - apply and_intro. by rewrite bare_elim sep_elim_l. by rewrite sep_elim_r.
Qed.
Lemma persistently_and_bare_sep_r P Q : P   Q  P   Q.
Proof. by rewrite !(comm _ P) persistently_and_bare_sep_l. Qed.
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Lemma and_sep_bare_persistently P Q :  P