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From iris.base_logic Require Export primitive.
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Set Default Proof Using "Type".
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Import upred.uPred primitive.uPred.
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Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P  Q)  (Q  P))%I.
Instance: Params (@uPred_iff) 1.
Infix "↔" := uPred_iff : uPred_scope.

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Definition uPred_laterN {M} (n : nat) (P : uPred M) : uPred M :=
  Nat.iter n uPred_later P.
Instance: Params (@uPred_laterN) 2.
Notation "▷^ n P" := (uPred_laterN n P)
  (at level 20, n at level 9, P at level 20,
   format "▷^ n  P") : uPred_scope.
Notation "▷? p P" := (uPred_laterN (Nat.b2n p) P)
  (at level 20, p at level 9, P at level 20,
   format "▷? p  P") : uPred_scope.

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Definition uPred_persistently_if {M} (p : bool) (P : uPred M) : uPred M :=
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  (if p then  P else P)%I.
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Instance: Params (@uPred_persistently_if) 2.
Arguments uPred_persistently_if _ !_ _/.
Notation "□? p P" := (uPred_persistently_if p P)
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  (at level 20, p at level 9, P at level 20, format "□? p  P").
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Definition uPred_except_0 {M} (P : uPred M) : uPred M :=  False  P.
Notation "◇ P" := (uPred_except_0 P)
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  (at level 20, right associativity) : uPred_scope.
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Instance: Params (@uPred_except_0) 1.
Typeclasses Opaque uPred_except_0.
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Class Timeless {M} (P : uPred M) := timelessP :  P   P.
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Arguments timelessP {_} _ {_}.
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Hint Mode Timeless + ! : typeclass_instances.
Instance: Params (@Timeless) 1.
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Class Persistent {M} (P : uPred M) := persistent : P   P.
Arguments persistent {_} _ {_}.
Hint Mode Persistent + ! : typeclass_instances.
Instance: Params (@Persistent) 1.
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Class Plain {M} (P : uPred M) := plain : P   P.
Arguments plain {_} _ {_}.
Hint Mode Plain + ! : typeclass_instances.
Instance: Params (@Plain) 1.

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Module uPred.
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Section derived.
Context {M : ucmraT}.
Implicit Types φ : Prop.
Implicit Types P Q : uPred M.
Implicit Types A : Type.
Notation "P ⊢ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *)
Notation "P ⊣⊢ Q" := (equiv (A:=uPred M) P%I Q%I). (* Force implicit argument M *)

(* Derived logical stuff *)
Lemma False_elim P : False  P.
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Proof. by apply (pure_elim' False). Qed.
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Lemma True_intro P : P  True.
Proof. by apply pure_intro. Qed.

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  uPred M) a : (P  Ψ a)  P   a, Ψ a.
Proof. intros ->; apply exist_intro. Qed.
Lemma forall_elim' {A} P (Ψ : A  uPred M) : (P   a, Ψ a)   a, P  Ψ a.
Proof. move=> HP a. by rewrite HP forall_elim. Qed.

Hint Resolve pure_intro.
Hint Resolve or_elim or_intro_l' or_intro_r'.
Hint Resolve and_intro and_elim_l' and_elim_r'.
Hint Immediate True_intro False_elim.

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_l P Q : (P  Q)  P  Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_r P Q : P  (P  Q)  Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_l' P Q R : (P  Q  R)  P  Q  R.
Proof. intros; apply impl_elim with Q; auto. Qed.
Lemma impl_elim_r' P Q R : (Q  P  R)  P  Q  R.
Proof. intros; apply impl_elim with P; auto. Qed.
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Lemma impl_entails P Q : (P  Q)%I  P  Q.
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Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed.
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Lemma entails_impl P Q : (P  Q)  (P  Q)%I.
Proof. intro. apply impl_intro_l. auto. 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  uPred M) :
  ( 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  uPred M) :
  ( 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 (() ==> () ==> ()) (@uPred_and M).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance and_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_and M).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance or_mono' : Proper (() ==> () ==> ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance or_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance impl_mono' :
  Proper (flip () ==> () ==> ()) (@uPred_impl M).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
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Global Instance impl_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@uPred_impl M).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
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Global Instance forall_mono' A :
  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
Proof. intros P1 P2; apply forall_mono. Qed.
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Global Instance forall_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@uPred_forall M A).
Proof. intros P1 P2; apply forall_mono. Qed.
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Global Instance exist_mono' A :
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  Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
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Proof. intros P1 P2; apply exist_mono. Qed.
Global Instance exist_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@uPred_exist M A).
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Proof. intros P1 P2; apply exist_mono. Qed.

Global Instance and_idem : IdemP () (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_idem : IdemP () (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_comm : Comm () (@uPred_and M).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance True_and : LeftId () True%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_True : RightId () True%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_and : LeftAbsorb () False%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_False : RightAbsorb () False%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance True_or : LeftAbsorb () True%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_True : RightAbsorb () True%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_or : LeftId () False%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_False : RightId () False%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_assoc : Assoc () (@uPred_and M).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance or_comm : Comm () (@uPred_or M).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance or_assoc : Assoc () (@uPred_or M).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance True_impl : LeftId () True%I (@uPred_impl M).
Proof.
  intros P; apply (anti_symm ()).
  - by rewrite -(left_id True%I uPred_and (_  _)%I) impl_elim_r.
  - by apply impl_intro_l; rewrite left_id.
Qed.
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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 exists_impl_forall {A} P (Ψ : A  uPred M) :
  (( 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  uPred M) : 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  uPred M) : ( a, Φ a)  P   a, Φ a  P.
Proof.
  rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm.
Qed.
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Lemma or_exist {A} (Φ Ψ : A  uPred M) :
  ( 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.
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Lemma pure_elim φ Q R : (Q  ⌜φ⌝)  (φ  Q  R)  Q  R.
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Proof.
  intros HQ HQR. rewrite -(idemp uPred_and Q) {1}HQ.
  apply impl_elim_l', pure_elim'=> ?. by apply entails_impl, HQR.
Qed.
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Lemma pure_mono φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
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Proof. intros; apply pure_elim with φ1; eauto. Qed.
Global Instance pure_mono' : Proper (impl ==> ()) (@uPred_pure M).
Proof. intros φ1 φ2; apply pure_mono. Qed.
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Global Instance pure_flip_mono : Proper (flip impl ==> flip ()) (@uPred_pure M).
Proof. intros φ1 φ2; apply pure_mono. Qed.
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Lemma pure_iff φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
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Proof. intros [??]; apply (anti_symm _); auto using pure_mono. Qed.
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Lemma pure_intro_l φ Q R : φ  (⌜φ⌝  Q  R)  Q  R.
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Proof. intros ? <-; auto using pure_intro. Qed.
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Lemma pure_intro_r φ Q R : φ  (Q  ⌜φ⌝  R)  Q  R.
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Proof. intros ? <-; auto. Qed.
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Lemma pure_intro_impl φ Q R : φ  (Q  ⌜φ⌝  R)  Q  R.
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Proof. intros ? ->. eauto using pure_intro_l, impl_elim_r. Qed.
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Lemma pure_elim_l φ Q R : (φ  Q  R)  ⌜φ⌝  Q  R.
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Proof. intros; apply pure_elim with φ; eauto. Qed.
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Lemma pure_elim_r φ Q R : (φ  Q  R)  Q  ⌜φ⌝  R.
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Proof. intros; apply pure_elim with φ; eauto. Qed.
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Lemma pure_True (φ : Prop) : φ  ⌜φ⌝  True.
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Proof. intros; apply (anti_symm _); auto. Qed.
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Lemma pure_False (φ : Prop) : ¬φ  ⌜φ⌝  False.
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Proof. intros; apply (anti_symm _); eauto using pure_elim. Qed.
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Lemma pure_and φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
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Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[??]; auto.
  - eapply (pure_elim φ1); [auto|]=> ?. eapply (pure_elim φ2); auto.
Qed.
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Lemma pure_or φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
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Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[?|?]; auto.
  - apply or_elim; eapply pure_elim; eauto.
Qed.
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Lemma pure_impl φ1 φ2 : ⌜φ1  φ2  (⌜φ1  ⌜φ2).
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Proof.
  apply (anti_symm _).
  - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver.
  - rewrite -pure_forall_2. apply forall_intro=> ?.
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    by rewrite -(left_id True uPred_and (_→_))%I (pure_True φ1) // impl_elim_r.
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Qed.
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Lemma pure_forall {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
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Proof.
  apply (anti_symm _); auto using pure_forall_2.
  apply forall_intro=> x. eauto using pure_mono.
Qed.
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Lemma pure_exist {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
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Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[x ?]. rewrite -(exist_intro x); auto.
  - apply exist_elim=> x. eauto using pure_mono.
Qed.

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Lemma internal_eq_refl' {A : ofeT} (a : A) P : P  a  a.
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Proof. rewrite (True_intro P). apply internal_eq_refl. Qed.
Hint Resolve internal_eq_refl'.
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Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a  b  P  a  b.
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Proof. by intros ->. Qed.
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Lemma internal_eq_sym {A : ofeT} (a b : A) : a  b  b  a.
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Proof. apply (internal_eq_rewrite a b (λ b, b  a)%I); auto. solve_proper. Qed.
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Lemma internal_eq_rewrite_contractive {A : ofeT} a b (Ψ : A  uPred M) P
  {HΨ : Contractive Ψ} : (P   (a  b))  (P  Ψ a)  P  Ψ b.
Proof.
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  move: HΨ=> /contractiveI HΨ Heq ?.
  apply (internal_eq_rewrite (Ψ a) (Ψ b) id _)=>//=. by rewrite -HΨ.
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Qed.
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Lemma pure_impl_forall φ P : (⌜φ⌝  P)  ( _ : φ, P).
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Proof.
  apply (anti_symm _).
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  - apply forall_intro=> ?. by rewrite pure_True // left_id.
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  - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ).
Qed.
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Lemma pure_alt φ : ⌜φ⌝   _ : φ, True.
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Proof.
  apply (anti_symm _).
  - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto.
  - by apply exist_elim, pure_intro.
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.
  apply and_intro. by rewrite (forall_elim true). by 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.
  apply or_elim. by rewrite -(exist_intro true). by rewrite -(exist_intro false).
Qed.

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

Lemma iff_refl Q P : Q  P  P.
Proof. rewrite /uPred_iff; apply and_intro; apply impl_intro_l; auto. Qed.
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Lemma iff_equiv P Q : (P  Q)%I  (P  Q).
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Proof.
  intros HPQ; apply (anti_symm ());
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    apply impl_entails; rewrite /uPred_valid HPQ /uPred_iff; auto.
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Qed.
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Lemma equiv_iff P Q : (P  Q)  (P  Q)%I.
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Proof. intros ->; apply iff_refl. Qed.
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Lemma internal_eq_iff P Q : P  Q  P  Q.
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Proof.
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  apply (internal_eq_rewrite P Q (λ Q, P  Q))%I;
    first solve_proper; auto using iff_refl.
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Qed.

(* Derived BI Stuff *)
Hint Resolve sep_mono.
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Lemma sep_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
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Proof. by intros; apply sep_mono. Qed.
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Lemma sep_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
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Proof. by apply sep_mono. Qed.
Global Instance sep_mono' : Proper (() ==> () ==> ()) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Global Instance sep_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
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Lemma wand_mono P P' Q Q' : (Q  P)  (P'  Q')  (P - P')  Q - Q'.
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Proof.
  intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'.
Qed.
Global Instance wand_mono' : Proper (flip () ==> () ==> ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
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Global Instance wand_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
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Global Instance sep_comm : Comm () (@uPred_sep M).
Proof. intros P Q; apply (anti_symm _); auto using sep_comm'. Qed.
Global Instance sep_assoc : Assoc () (@uPred_sep M).
Proof.
  intros P Q R; apply (anti_symm _); auto using sep_assoc'.
  by rewrite !(comm _ P) !(comm _ _ R) sep_assoc'.
Qed.
Global Instance True_sep : LeftId () True%I (@uPred_sep M).
Proof. intros P; apply (anti_symm _); auto using True_sep_1, True_sep_2. Qed.
Global Instance sep_True : RightId () True%I (@uPred_sep M).
Proof. by intros P; rewrite comm left_id. Qed.
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Lemma sep_elim_l P Q : P  Q  P.
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Proof. by rewrite (True_intro Q) right_id. Qed.
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Lemma sep_elim_r P Q : P  Q  Q.
Proof. by rewrite (comm ())%I; apply sep_elim_l. Qed.
Lemma sep_elim_l' P Q R : (P  R)  P  Q  R.
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Proof. intros ->; apply sep_elim_l. Qed.
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Lemma sep_elim_r' P Q R : (Q  R)  P  Q  R.
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Proof. intros ->; apply sep_elim_r. Qed.
Hint Resolve sep_elim_l' sep_elim_r'.
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Lemma sep_intro_True_l P Q R : P%I  (R  Q)  R  P  Q.
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Proof. by intros; rewrite -(left_id True%I uPred_sep R); apply sep_mono. Qed.
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Lemma sep_intro_True_r P Q R : (R  P)  Q%I  R  P  Q.
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Proof. by intros; rewrite -(right_id True%I uPred_sep R); apply sep_mono. Qed.
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Lemma sep_elim_True_l P Q R : P  (P  R  Q)  R  Q.
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Proof. by intros HP; rewrite -HP left_id. Qed.
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Lemma sep_elim_True_r P Q R : P  (R  P  Q)  R  Q.
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Proof. by intros HP; rewrite -HP right_id. Qed.
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Lemma wand_intro_l P Q R : (Q  P  R)  P  Q - R.
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Proof. rewrite comm; apply wand_intro_r. Qed.
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Lemma wand_elim_l P Q : (P - Q)  P  Q.
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Proof. by apply wand_elim_l'. Qed.
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Lemma wand_elim_r P Q : P  (P - Q)  Q.
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Proof. rewrite (comm _ P); apply wand_elim_l. Qed.
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Lemma wand_elim_r' P Q R : (Q  P - R)  P  Q  R.
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Proof. intros ->; apply wand_elim_r. Qed.
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Lemma wand_apply P Q R S : (P  Q - R)  (S  P  Q)  S  R.
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Proof. intros HR%wand_elim_l' HQ. by rewrite HQ. Qed.
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Lemma wand_frame_l P Q R : (Q - R)  P  Q - P  R.
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Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed.
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Lemma wand_frame_r P Q R : (Q - R)  Q  P - R  P.
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Proof.
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  apply wand_intro_l. rewrite ![(_  P)%I]comm -assoc.
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  apply sep_mono_r, wand_elim_r.
Qed.
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Lemma wand_diag P : (P - P)  True.
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Proof. apply (anti_symm _); auto. apply wand_intro_l; by rewrite right_id. Qed.
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Lemma wand_True P : (True - P)  P.
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Proof.
  apply (anti_symm _); last by auto using wand_intro_l.
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  eapply sep_elim_True_l; last by apply wand_elim_r. done.
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Qed.
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Lemma wand_entails P Q : (P - Q)%I  P  Q.
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Proof.
  intros HPQ. eapply sep_elim_True_r; first exact: HPQ. by rewrite wand_elim_r.
Qed.
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Lemma entails_wand P Q : (P  Q)  (P - Q)%I.
Proof. intro. apply wand_intro_l. auto. Qed.
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Lemma wand_curry P Q R : (P - Q - R)  (P  Q - R).
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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.

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Lemma sep_and P Q : (P  Q)  (P  Q).
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Proof. auto. Qed.
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Lemma impl_wand_1 P Q : (P  Q)  P - Q.
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Proof. apply wand_intro_r, impl_elim with P; auto. Qed.
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Lemma pure_elim_sep_l φ Q R : (φ  Q  R)  ⌜φ⌝  Q  R.
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Proof. intros; apply pure_elim with φ; eauto. Qed.
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Lemma pure_elim_sep_r φ Q R : (φ  Q  R)  Q  ⌜φ⌝  R.
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Proof. intros; apply pure_elim with φ; eauto. Qed.

Global Instance sep_False : LeftAbsorb () False%I (@uPred_sep M).
Proof. intros P; apply (anti_symm _); auto. Qed.
Global Instance False_sep : RightAbsorb () False%I (@uPred_sep M).
Proof. intros P; apply (anti_symm _); auto. Qed.

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

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(* Plainness modality *)
Global Instance plainly_mono' : Proper (() ==> ()) (@uPred_plainly M).
Proof. intros P Q; apply plainly_mono. Qed.
Global Instance naugth_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_plainly M).
Proof. intros P Q; apply plainly_mono. Qed.

Lemma plainly_elim P :  P  P.
Proof. by rewrite plainly_elim' persistently_elim. Qed.
Hint Resolve plainly_mono plainly_elim.
Lemma plainly_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. apply plainly_idemp. Qed.
Lemma plainly_idemp P :   P   P.
Proof. apply (anti_symm _); auto using plainly_idemp. Qed.

Lemma persistently_plainly P :   P   P.
Proof.
  apply (anti_symm _); auto using persistently_elim.
  by rewrite -plainly_elim' plainly_idemp.
Qed.
Lemma plainly_persistently P :   P   P.
Proof.
  apply (anti_symm _); auto using plainly_mono, persistently_elim.
  by rewrite -plainly_elim' plainly_idemp.
Qed.

Lemma plainly_pure φ :  ⌜φ⌝  ⌜φ⌝.
Proof.
  apply (anti_symm _); auto.
  apply pure_elim'=> Hφ.
  trans ( x : False,  True : uPred M)%I; [by apply forall_intro|].
  rewrite plainly_forall_2. auto using plainly_mono, pure_intro.
Qed.
Lemma plainly_forall {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
Proof.
  apply (anti_symm _); auto using plainly_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
Lemma plainly_exist {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
Proof.
  apply (anti_symm _); auto using plainly_exist_1.
  apply exist_elim=> x. by rewrite (exist_intro x).
Qed.
Lemma plainly_and P Q :  (P  Q)   P   Q.
Proof. rewrite !and_alt plainly_forall. by apply forall_proper=> -[]. Qed.
Lemma plainly_or P Q :  (P  Q)   P   Q.
Proof. rewrite !or_alt plainly_exist. by apply exist_proper=> -[]. Qed.
Lemma plainly_impl P Q :  (P  Q)   P   Q.
Proof.
  apply impl_intro_l; rewrite -plainly_and.
  apply plainly_mono, impl_elim with P; auto.
Qed.
Lemma plainly_internal_eq {A:ofeT} (a b : A) :  (a  b)  a  b.
Proof.
  apply (anti_symm ()); auto using persistently_elim.
  apply (internal_eq_rewrite a b (λ b,  (a  b))%I); auto.
  { intros n; solve_proper. }
  rewrite -(internal_eq_refl a) plainly_pure; auto.
Qed.

Lemma plainly_and_sep_l_1 P Q :  P  Q   P  Q.
Proof. by rewrite -persistently_plainly persistently_and_sep_l_1. Qed.
Lemma plainly_and_sep_l' P Q :  P  Q   P  Q.
Proof. apply (anti_symm ()); auto using plainly_and_sep_l_1. Qed.
Lemma plainly_and_sep_r' P Q : P   Q  P   Q.
Proof. by rewrite !(comm _ P) plainly_and_sep_l'. Qed.
Lemma plainly_sep_dup' P :  P   P   P.
Proof. by rewrite -plainly_and_sep_l' idemp. Qed.

Lemma plainly_and_sep P Q :  (P  Q)   (P  Q).
Proof.
  apply (anti_symm ()); auto.
  rewrite -{1}plainly_idemp plainly_and plainly_and_sep_l'; auto.
Qed.
Lemma plainly_sep P Q :  (P  Q)   P   Q.
Proof. by rewrite -plainly_and_sep -plainly_and_sep_l' plainly_and. Qed.

Lemma plainly_wand P Q :  (P - Q)   P -  Q.
Proof. by apply wand_intro_r; rewrite -plainly_sep wand_elim_l. Qed.
Lemma plainly_impl_wand P Q :  (P  Q)   (P - Q).
Proof.
  apply (anti_symm ()); [by rewrite -impl_wand_1|].
  apply plainly_intro', impl_intro_r.
  by rewrite plainly_and_sep_l' plainly_elim wand_elim_l.
Qed.
Lemma wand_impl_plainly P Q : ( P - Q)  ( P  Q).
Proof.
  apply (anti_symm ()); [|by rewrite -impl_wand_1].
  apply impl_intro_l. by rewrite plainly_and_sep_l' wand_elim_r.
Qed.
Lemma plainly_entails_l' P Q : (P   Q)  P   Q  P.
Proof. intros; rewrite -plainly_and_sep_l'; auto. Qed.
Lemma plainly_entails_r' P Q : (P   Q)  P  P   Q.
Proof. intros; rewrite -plainly_and_sep_r'; auto. Qed.

Lemma plainly_laterN n P :  ^n P  ^n  P.
Proof. induction n as [|n IH]; simpl; auto. by rewrite plainly_later IH. Qed.

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(* Always derived *)
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Hint Resolve persistently_mono persistently_elim.
Global Instance persistently_mono' : Proper (() ==> ()) (@uPred_persistently M).
Proof. intros P Q; apply persistently_mono. Qed.
Global Instance persistently_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_persistently M).
Proof. intros P Q; apply persistently_mono. Qed.
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Lemma persistently_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. apply persistently_idemp_2. Qed.
Lemma persistently_idemp P :   P   P.
Proof. apply (anti_symm _); auto using persistently_idemp_2. Qed.
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Lemma persistently_pure φ :  ⌜φ⌝  ⌜φ⌝.
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Proof. by rewrite -plainly_pure persistently_plainly. Qed.
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Lemma persistently_forall {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
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Proof.
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  apply (anti_symm _); auto using persistently_forall_2.
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  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
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Lemma persistently_exist {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
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Proof.
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  apply (anti_symm _); auto using persistently_exist_1.
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  apply exist_elim=> x. by rewrite (exist_intro x).
Qed.
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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.
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Proof.
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  apply impl_intro_l; rewrite -persistently_and.
  apply persistently_mono, impl_elim with P; auto.
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Qed.
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Lemma persistently_internal_eq {A:ofeT} (a b : A) :  (a  b)  a  b.
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Proof. by rewrite -plainly_internal_eq persistently_plainly. Qed.
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Lemma persistently_and_sep_l P Q :  P  Q   P  Q.
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Proof. apply (anti_symm ()); auto using persistently_and_sep_l_1. Qed.
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Lemma persistently_and_sep_r P Q : P   Q  P   Q.
Proof. by rewrite !(comm _ P) persistently_and_sep_l. Qed.
Lemma persistently_sep_dup P :  P   P   P.
Proof. by rewrite -persistently_and_sep_l idemp. Qed.
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Lemma persistently_and_sep P Q :  (P  Q)   (P  Q).
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Proof.
  apply (anti_symm ()); auto.
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  rewrite -{1}persistently_idemp persistently_and persistently_and_sep_l; auto.
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Qed.
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Lemma persistently_sep P Q :  (P  Q)   P   Q.
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Proof. by rewrite -persistently_and_sep -persistently_and_sep_l persistently_and. Qed.
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Lemma persistently_wand P Q :  (P - Q)   P -  Q.
Proof. by apply wand_intro_r; rewrite -persistently_sep wand_elim_l. Qed.
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Lemma persistently_impl_wand P Q :  (P  Q)   (P - Q).
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Proof.
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  apply (anti_symm ()); [by rewrite -impl_wand_1|].
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  apply persistently_intro', impl_intro_r.
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  by rewrite persistently_and_sep_l persistently_elim wand_elim_l.
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Qed.
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Lemma impl_wand_persistently P Q : ( P  Q)  ( P - Q).
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Proof.
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  apply (anti_symm ()); [by rewrite -impl_wand_1|].
  apply impl_intro_l. by rewrite persistently_and_sep_l wand_elim_r.
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Qed.
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Lemma persistently_entails_l P Q : (P   Q)  P   Q  P.
Proof. intros; rewrite -persistently_and_sep_l; auto. Qed.
Lemma persistently_entails_r P Q : (P   Q)  P  P   Q.
Proof. intros; rewrite -persistently_and_sep_r; auto. Qed.
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Lemma persistently_laterN n P :  ^n P  ^n  P.
Proof. induction n as [|n IH]; simpl; auto. by rewrite persistently_later IH. Qed.
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Lemma wand_alt P Q : (P - Q)   R, R   (P  R  Q).
Proof.
  apply (anti_symm ()).
  - rewrite -(right_id True%I uPred_sep (P - Q)%I) -(exist_intro (P - Q)%I).
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    apply sep_mono_r. rewrite -persistently_pure. apply persistently_mono, impl_intro_l.
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    by rewrite wand_elim_r 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 uPred_and (P  Q)%I) -(exist_intro (P  Q)%I).
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    apply and_mono_r. rewrite -persistently_pure. apply persistently_mono, wand_intro_l.
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    by rewrite impl_elim_r right_id.
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  - apply exist_elim=> R. apply impl_intro_l. rewrite assoc persistently_and_sep_r.
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    by rewrite persistently_elim wand_elim_r.
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Qed.
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(* Later derived *)
Lemma later_proper P Q : (P  Q)   P   Q.
Proof. by intros ->. Qed.
Hint Resolve later_mono later_proper.
Global Instance later_mono' : Proper (() ==> ()) (@uPred_later M).
Proof. intros P Q; apply later_mono. Qed.
Global Instance later_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_later M).
Proof. intros P Q; apply later_mono. Qed.

Lemma later_intro P : P   P.
Proof.
  rewrite -(and_elim_l ( P) P) -(löb ( P  P)).
  apply impl_intro_l. by rewrite {1}(and_elim_r ( P)).
Qed.

Lemma later_True :  True  True.
Proof. apply (anti_symm ()); auto using later_intro. Qed.
Lemma later_forall {A} (Φ : A  uPred M) : (  a, Φ a)  ( a,  Φ a).
Proof.
  apply (anti_symm _); auto using later_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
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Lemma later_exist_2 {A} (Φ : A  uPred M) : ( a,  Φ a)   ( a, Φ a).
Proof. apply exist_elim; eauto using exist_intro. Qed.
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Lemma later_exist `{Inhabited A} (Φ : A  uPred M) :
   ( a, Φ a)  ( a,  Φ a).
Proof.
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  apply: anti_symm; [|apply later_exist_2].
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  rewrite later_exist_false. apply or_elim; last done.
  rewrite -(exist_intro inhabitant); auto.
Qed.
Lemma later_and P Q :  (P  Q)   P   Q.
Proof. rewrite !and_alt later_forall. by apply forall_proper=> -[]. Qed.
Lemma later_or P Q :  (P  Q)   P   Q.
Proof. rewrite !or_alt later_exist. by apply exist_proper=> -[]. Qed.
Lemma later_impl P Q :  (P  Q)   P   Q.
Proof. apply impl_intro_l; rewrite -later_and; eauto using impl_elim. Qed.
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Lemma later_wand P Q :  (P - Q)   P -  Q.
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Proof. apply wand_intro_r; rewrite -later_sep; eauto using wand_elim_l. Qed.
Lemma later_iff P Q :  (P  Q)   P   Q.
Proof. by rewrite /uPred_iff later_and !later_impl. Qed.

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(* Iterated later modality *)
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Global Instance laterN_ne m : NonExpansive (@uPred_laterN M m).
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Proof. induction m; simpl. by intros ???. solve_proper. Qed.
Global Instance laterN_proper m :
  Proper (() ==> ()) (@uPred_laterN M m) := ne_proper _.

Lemma laterN_0 P : ^0 P  P.
Proof. done. Qed.
Lemma later_laterN n P : ^(S n) P   ^n P.
Proof. done. Qed.
Lemma laterN_later n P : ^(S n) P  ^n  P.
Proof. induction n; simpl; auto. Qed.
Lemma laterN_plus n1 n2 P : ^(n1 + n2) P  ^n1 ^n2 P.
Proof. induction n1; simpl; auto. Qed.
Lemma laterN_le n1 n2 P : n1  n2  ^n1 P  ^n2 P.
Proof. induction 1; simpl; by rewrite -?later_intro. Qed.

Lemma laterN_mono n P Q : (P  Q)  ^n P  ^n Q.
Proof. induction n; simpl; auto. Qed.
Global Instance laterN_mono' n : Proper (() ==> ()) (@uPred_laterN M n).
Proof. intros P Q; apply laterN_mono. Qed.
Global Instance laterN_flip_mono' n :
  Proper (flip () ==> flip ()) (@uPred_laterN M n).
Proof. intros P Q; apply laterN_mono. Qed.

Lemma laterN_intro n P : P  ^n P.
Proof. induction n as [|n IH]; simpl; by rewrite -?later_intro. Qed.

Lemma laterN_True n : ^n True  True.
Proof. apply (anti_symm ()); auto using laterN_intro. Qed.
Lemma laterN_forall {A} n (Φ : A  uPred M) : (^n  a, Φ a)  ( a, ^n Φ a).
Proof. induction n as [|n IH]; simpl; rewrite -?later_forall; auto. Qed.
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Lemma laterN_exist_2 {A} n (Φ : A  uPred M) : ( a, ^n Φ a)  ^n ( a, Φ a).
Proof. apply exist_elim; eauto using exist_intro, laterN_mono. Qed.
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Lemma laterN_exist `{Inhabited A} n (Φ : A  uPred M) :
  (^n  a, Φ a)   a, ^n Φ a.
Proof. induction n as [|n IH]; simpl; rewrite -?later_exist; auto. Qed.
Lemma laterN_and n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. induction n as [|n IH]; simpl; rewrite -?later_and; auto. Qed.
Lemma laterN_or n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. induction n as [|n IH]; simpl; rewrite -?later_or; auto. Qed.
Lemma laterN_impl n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof.
  apply impl_intro_l; rewrite -laterN_and; eauto using impl_elim, laterN_mono.
Qed.
Lemma laterN_sep n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. induction n as [|n IH]; simpl; rewrite -?later_sep; auto. Qed.
Lemma laterN_wand n P Q : ^n (P - Q)  ^n P - ^n Q.
Proof.
  apply wand_intro_r; rewrite -laterN_sep; eauto using wand_elim_l,laterN_mono.
Qed.
Lemma laterN_iff n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. by rewrite /uPred_iff laterN_and !laterN_impl. Qed.

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(* Conditional persistently *)
Global Instance persistently_if_ne p : NonExpansive (@uPred_persistently_if M p).
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Proof. solve_proper. Qed.
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Global Instance persistently_if_proper p : Proper (() ==> ()) (@uPred_persistently_if M p).
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Proof. solve_proper. Qed.
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Global Instance persistently_if_mono p : Proper (() ==> ()) (@uPred_persistently_if M p).
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Proof. solve_proper. Qed.

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Lemma persistently_if_elim p P : ?p P  P.
Proof. destruct p; simpl; auto using persistently_elim. Qed.
Lemma persistently_elim_if p P :  P  ?p P.
Proof. destruct p; simpl; auto using persistently_elim. Qed.

Lemma persistently_if_pure p φ : ?p ⌜φ⌝  ⌜φ⌝.
Proof. destruct p; simpl; auto using persistently_pure. Qed.
Lemma persistently_if_and p P Q : ?p (P  Q)  ?p P  ?p Q.
Proof. destruct p; simpl; auto using persistently_and. Qed.
Lemma persistently_if_or p P Q : ?p (P  Q)  ?p P  ?p Q.
Proof. destruct p; simpl; auto using persistently_or. Qed.
Lemma persistently_if_exist {A} p (Ψ : A  uPred M) : (?p  a, Ψ a)   a, ?p Ψ a.
Proof. destruct p; simpl; auto using persistently_exist. Qed.
Lemma persistently_if_sep p P Q : ?p (P  Q)  ?p P  ?p Q.
Proof. destruct p; simpl; auto using persistently_sep. Qed.
Lemma persistently_if_later p P : ?p  P   ?p P.
Proof. destruct p; simpl; auto using persistently_later. Qed.
Lemma persistently_if_laterN p n P : ?p ^n P  ^n ?p P.
Proof. destruct p; simpl; auto using persistently_laterN. Qed.
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(* True now *)
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Global Instance except_0_ne : NonExpansive (@uPred_except_0 M).
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Proof. solve_proper. Qed.
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Global Instance except_0_proper : Proper (() ==> ()) (@uPred_except_0 M).
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Proof. solve_proper. Qed.
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Global Instance except_0_mono' : Proper (() ==> ()) (@uPred_except_0 M).
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Proof. solve_proper. Qed.
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Global Instance except_0_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_except_0 M).
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Proof. solve_proper. Qed.

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Lemma except_0_intro P : P   P.
Proof. rewrite /uPred_except_0; auto. Qed.
Lemma except_0_mono P Q : (P  Q)   P   Q.
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Proof. by intros ->. Qed.
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Lemma except_0_idemp P :   P   P.
Proof. rewrite /uPred_except_0; auto. Qed.

Lemma except_0_True :  True  True.
Proof. rewrite /uPred_except_0. apply (anti_symm _); auto. Qed.
Lemma except_0_or P Q :  (P  Q)   P   Q.
Proof. rewrite /uPred_except_0. apply (anti_symm _); auto. Qed.
Lemma except_0_and P Q :  (P  Q)   P   Q.
Proof. by rewrite /uPred_except_0 or_and_l. Qed.
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Lemma except_0_sep P Q :  (P  Q)   P   Q.
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Proof.
  rewrite /uPred_except_0. apply (anti_symm _).
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  - apply or_elim; last by auto.
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    by rewrite -!or_intro_l -persistently_pure -persistently_later -persistently_sep_dup.
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  - rewrite sep_or_r sep_elim_l sep_or_l; auto.
Qed.
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Lemma except_0_forall {A} (Φ : A  uPred M) :  ( a, Φ a)   a,  Φ a.
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Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
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Lemma except_0_exist_2 {A} (Φ : A  uPred M) : ( a,  Φ a)    a, Φ a.
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Proof. apply exist_elim=> a. by rewrite (exist_intro