Library iris.bi.plainly

From iris.bi Require Import derived_laws_later big_op internal_eq.
From iris.algebra Require Import monoid.
Import interface.bi derived_laws.bi derived_laws_later.bi.

Class Plainly (A : Type) := plainly : A A.
Arguments plainly {A}%type_scope {_} _%I.
Hint Mode Plainly ! : typeclass_instances.
Instance: Params (@plainly) 2 := {}.
Notation "■ P" := (plainly P) : bi_scope.

Record BiPlainlyMixin (PROP : bi) `(Plainly PROP) := {
  bi_plainly_mixin_plainly_ne : NonExpansive (plainly (A:=PROP));

  bi_plainly_mixin_plainly_mono (P Q : PROP) : (P Q) P Q;
  bi_plainly_mixin_plainly_elim_persistently (P : PROP) : P <pers> P;
  bi_plainly_mixin_plainly_idemp_2 (P : PROP) : P P;

  bi_plainly_mixin_plainly_forall_2 {A} (Ψ : A PROP) :
    ( a, (Ψ a)) ( a, Ψ a);

  
  bi_plainly_mixin_persistently_impl_plainly (P Q : PROP) :
    ( P <pers> Q) <pers> ( P Q);
  bi_plainly_mixin_plainly_impl_plainly (P Q : PROP) :
    ( P Q) (■ P Q);

  bi_plainly_mixin_plainly_emp_intro (P : PROP) : P emp;
  bi_plainly_mixin_plainly_absorb (P Q : PROP) : P Q P;

  bi_plainly_mixin_later_plainly_1 (P : PROP) : P P;
  bi_plainly_mixin_later_plainly_2 (P : PROP) : P P;
}.

Class BiPlainly (PROP : bi) := {
  bi_plainly_plainly :> Plainly PROP;
  bi_plainly_mixin : BiPlainlyMixin PROP bi_plainly_plainly;
}.
Hint Mode BiPlainly ! : typeclass_instances.
Arguments bi_plainly_plainly : simpl never.

Class BiPlainlyExist `{!BiPlainly PROP} :=
  plainly_exist_1 A (Ψ : A PROP) :
     ( a, Ψ a) a, (Ψ a).
Arguments BiPlainlyExist : clear implicits.
Arguments BiPlainlyExist _ {_}.
Arguments plainly_exist_1 _ {_ _} _.
Hint Mode BiPlainlyExist ! - : typeclass_instances.

Class BiPropExt `{!BiPlainly PROP, !BiInternalEq PROP} :=
  prop_ext_2 (P Q : PROP) : (P ∗-∗ Q) P Q.
Arguments BiPropExt : clear implicits.
Arguments BiPropExt _ {_ _}.
Arguments prop_ext_2 _ {_ _ _} _.
Hint Mode BiPropExt ! - - : typeclass_instances.

Section plainly_laws.
  Context `{BiPlainly PROP}.
  Implicit Types P Q : PROP.

  Global Instance plainly_ne : NonExpansive (@plainly PROP _).
  Proof. eapply bi_plainly_mixin_plainly_ne, bi_plainly_mixin. Qed.

  Lemma plainly_mono P Q : (P Q) P Q.
  Proof. eapply bi_plainly_mixin_plainly_mono, bi_plainly_mixin. Qed.
  Lemma plainly_elim_persistently P : P <pers> P.
  Proof. eapply bi_plainly_mixin_plainly_elim_persistently, bi_plainly_mixin. Qed.
  Lemma plainly_idemp_2 P : P P.
  Proof. eapply bi_plainly_mixin_plainly_idemp_2, bi_plainly_mixin. Qed.
  Lemma plainly_forall_2 {A} (Ψ : A PROP) : ( a, (Ψ a)) ( a, Ψ a).
  Proof. eapply bi_plainly_mixin_plainly_forall_2, bi_plainly_mixin. Qed.
  Lemma persistently_impl_plainly P Q : ( P <pers> Q) <pers> ( P Q).
  Proof. eapply bi_plainly_mixin_persistently_impl_plainly, bi_plainly_mixin. Qed.
  Lemma plainly_impl_plainly P Q : ( P Q) (■ P Q).
  Proof. eapply bi_plainly_mixin_plainly_impl_plainly, bi_plainly_mixin. Qed.
  Lemma plainly_absorb P Q : P Q P.
  Proof. eapply bi_plainly_mixin_plainly_absorb, bi_plainly_mixin. Qed.
  Lemma plainly_emp_intro P : P emp.
  Proof. eapply bi_plainly_mixin_plainly_emp_intro, bi_plainly_mixin. Qed.

  Lemma later_plainly_1 P : P ( P).
  Proof. eapply bi_plainly_mixin_later_plainly_1, bi_plainly_mixin. Qed.
  Lemma later_plainly_2 P : P P.
  Proof. eapply bi_plainly_mixin_later_plainly_2, bi_plainly_mixin. Qed.
End plainly_laws.

Class Plain `{BiPlainly PROP} (P : PROP) := plain : P P.
Arguments Plain {_ _} _%I : simpl never.
Arguments plain {_ _} _%I {_}.
Hint Mode Plain + - ! : typeclass_instances.
Instance: Params (@Plain) 1 := {}.

Definition plainly_if `{!BiPlainly PROP} (p : bool) (P : PROP) : PROP :=
  (if p then P else P)%I.
Arguments plainly_if {_ _} !_ _%I /.
Instance: Params (@plainly_if) 2 := {}.
Typeclasses Opaque plainly_if.

Notation "■? p P" := (plainly_if p P) : bi_scope.

Section plainly_derived.
Context `{BiPlainly PROP}.
Implicit Types P : PROP.

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

Global Instance plainly_proper :
  Proper ((⊣⊢) ==> (⊣⊢)) (@plainly PROP _) := ne_proper _.

Global Instance plainly_mono' : Proper ((⊢) ==> (⊢)) (@plainly PROP _).
Proof. intros P Q; apply plainly_mono. Qed.
Global Instance plainly_flip_mono' :
  Proper (flip (⊢) ==> flip (⊢)) (@plainly PROP _).
Proof. intros P Q; apply plainly_mono. Qed.

Lemma affinely_plainly_elim P : <affine> P P.
Proof. by rewrite plainly_elim_persistently /bi_affinely persistently_and_emp_elim. Qed.

Lemma persistently_elim_plainly P : <pers> P ⊣⊢ P.
Proof.
  apply (anti_symm _).
  - by rewrite persistently_into_absorbingly /bi_absorbingly comm plainly_absorb.
  - by rewrite {1}plainly_idemp_2 plainly_elim_persistently.
Qed.
Lemma persistently_if_elim_plainly P p : <pers>?p P ⊣⊢ P.
Proof. destruct p; last done. exact: persistently_elim_plainly. Qed.

Lemma plainly_persistently_elim P : <pers> P ⊣⊢ P.
Proof.
  apply (anti_symm _).
  - rewrite -{1}(left_id True%I bi_and ( _)%I) (plainly_emp_intro True%I).
    rewrite -{2}(persistently_and_emp_elim P).
    rewrite !and_alt -plainly_forall_2. by apply forall_mono⇒ -[].
  - by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
Qed.

Lemma absorbingly_elim_plainly P : <absorb> P ⊣⊢ P.
Proof. by rewrite -(persistently_elim_plainly P) absorbingly_elim_persistently. Qed.

Lemma plainly_and_sep_elim P Q : P Q -∗ (emp P) Q.
Proof. by rewrite plainly_elim_persistently persistently_and_sep_elim_emp. Qed.
Lemma plainly_and_sep_assoc P Q R : P (Q R) ⊣⊢ ( P Q) R.
Proof. by rewrite -(persistently_elim_plainly P) persistently_and_sep_assoc. Qed.
Lemma plainly_and_emp_elim P : emp P P.
Proof. by rewrite plainly_elim_persistently persistently_and_emp_elim. Qed.
Lemma plainly_into_absorbingly P : P <absorb> P.
Proof. by rewrite plainly_elim_persistently persistently_into_absorbingly. Qed.
Lemma plainly_elim P `{!Absorbing P} : P P.
Proof. by rewrite plainly_elim_persistently persistently_elim. Qed.

Lemma plainly_idemp_1 P : P P.
Proof. by rewrite plainly_into_absorbingly absorbingly_elim_plainly. Qed.
Lemma plainly_idemp P : P ⊣⊢ P.
Proof. apply (anti_symm _); auto using plainly_idemp_1, plainly_idemp_2. Qed.

Lemma plainly_intro' P Q : ( P Q) P Q.
Proof. intros <-. apply plainly_idemp_2. Qed.

Lemma plainly_pure φ : φ ⊣⊢@{PROP} φ.
Proof.
  apply (anti_symm _); auto.
  - by rewrite plainly_elim_persistently persistently_pure.
  - apply pure_elim'Hφ.
    trans ( x : False, True : PROP)%I; [by apply forall_intro|].
    rewrite plainly_forall_2. by rewrite -(pure_intro φ).
Qed.
Lemma plainly_forall {A} (Ψ : A PROP) : ( a, Ψ a) ⊣⊢ a, (Ψ a).
Proof.
  apply (anti_symm _); auto using plainly_forall_2.
  apply forall_introx. by rewrite (forall_elim x).
Qed.
Lemma plainly_exist_2 {A} (Ψ : A PROP) : ( a, (Ψ a)) ( a, Ψ a).
Proof. apply exist_elimx. by rewrite (exist_intro x). Qed.
Lemma plainly_exist `{!BiPlainlyExist PROP} {A} (Ψ : A PROP) :
   ( a, Ψ a) ⊣⊢ a, (Ψ a).
Proof. apply (anti_symm _); auto using plainly_exist_1, plainly_exist_2. Qed.
Lemma plainly_and P Q : (P Q) ⊣⊢ P Q.
Proof. rewrite !and_alt plainly_forall. by apply forall_proper⇒ -[]. Qed.
Lemma plainly_or_2 P Q : P Q (P Q).
Proof. rewrite !or_alt -plainly_exist_2. by apply exist_mono⇒ -[]. Qed.
Lemma plainly_or `{!BiPlainlyExist PROP} 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_emp_2 : emp ⊢@{PROP} emp.
Proof. apply plainly_emp_intro. Qed.

Lemma plainly_sep_dup P : P ⊣⊢ P P.
Proof.
  apply (anti_symm _).
  - rewrite -{1}(idemp bi_and ( _)%I).
    by rewrite -{2}(emp_sep ( _)%I) plainly_and_sep_assoc and_elim_l.
  - by rewrite plainly_absorb.
Qed.

Lemma plainly_and_sep_l_1 P Q : P Q P Q.
Proof. by rewrite -{1}(emp_sep Q%I) plainly_and_sep_assoc and_elim_l. Qed.
Lemma plainly_and_sep_r_1 P Q : P Q P Q.
Proof. by rewrite !(comm _ P) plainly_and_sep_l_1. Qed.

Lemma plainly_True_emp : True ⊣⊢@{PROP} emp.
Proof. apply (anti_symm _); eauto using plainly_mono, plainly_emp_intro. Qed.
Lemma plainly_and_sep P Q : (P Q) (P Q).
Proof.
  rewrite plainly_and.
  rewrite -{1}plainly_idemp -plainly_and -{1}(emp_sep Q%I).
  by rewrite plainly_and_sep_assoc (comm bi_and) plainly_and_emp_elim.
Qed.

Lemma plainly_affinely_elim P : <affine> P ⊣⊢ P.
Proof. by rewrite /bi_affinely plainly_and -plainly_True_emp plainly_pure left_id. Qed.

Lemma intuitionistically_plainly_elim P : P -∗ P.
Proof. rewrite intuitionistically_affinely plainly_elim_persistently //. Qed.
Lemma intuitionistically_plainly P : P -∗ P.
Proof.
  rewrite /bi_intuitionistically plainly_affinely_elim affinely_elim.
  rewrite persistently_elim_plainly plainly_persistently_elim. done.
Qed.

Lemma and_sep_plainly P Q : P Q ⊣⊢ P Q.
Proof.
  apply (anti_symm _); auto using plainly_and_sep_l_1.
  apply and_intro.
  - by rewrite plainly_absorb.
  - by rewrite comm plainly_absorb.
Qed.
Lemma plainly_sep_2 P Q : P Q (P Q).
Proof. by rewrite -plainly_and_sep plainly_and -and_sep_plainly. Qed.
Lemma plainly_sep `{BiPositive PROP} P Q : (P Q) ⊣⊢ P Q.
Proof.
  apply (anti_symm _); auto using plainly_sep_2.
  rewrite -(plainly_affinely_elim (_ _)%I) affinely_sep -and_sep_plainly. apply and_intro.
  - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
  - by rewrite (affinely_elim_emp P) left_id affinely_elim.
Qed.

Lemma plainly_wand P Q : (P -∗ Q) P -∗ Q.
Proof. apply wand_intro_r. by rewrite plainly_sep_2 wand_elim_l. Qed.

Lemma plainly_entails_l P Q : (P Q) P Q P.
Proof. intros; rewrite -plainly_and_sep_l_1; auto. Qed.
Lemma plainly_entails_r P Q : (P Q) P P Q.
Proof. intros; rewrite -plainly_and_sep_r_1; auto. Qed.

Lemma plainly_impl_wand_2 P Q : (P -∗ Q) (P Q).
Proof.
  apply plainly_intro', impl_intro_r.
  rewrite -{2}(emp_sep P%I) plainly_and_sep_assoc.
  by rewrite (comm bi_and) plainly_and_emp_elim wand_elim_l.
Qed.

Lemma impl_wand_plainly_2 P Q : ( P -∗ Q) ( P Q).
Proof. apply impl_intro_l. by rewrite plainly_and_sep_l_1 wand_elim_r. Qed.

Lemma impl_wand_affinely_plainly P Q : ( P Q) ⊣⊢ (<affine> P -∗ Q).
Proof. by rewrite -(persistently_elim_plainly P) impl_wand_intuitionistically. Qed.

Lemma persistently_wand_affinely_plainly P Q :
  (<affine> P -∗ <pers> Q) <pers> (<affine> P -∗ Q).
Proof. rewrite -!impl_wand_affinely_plainly. apply persistently_impl_plainly. Qed.

Lemma plainly_wand_affinely_plainly P Q :
  (<affine> P -∗ Q) (<affine> P -∗ Q).
Proof. rewrite -!impl_wand_affinely_plainly. apply plainly_impl_plainly. Qed.

Section plainly_affine_bi.
  Context `{BiAffine PROP}.

  Lemma plainly_emp : emp ⊣⊢@{PROP} emp.
  Proof. by rewrite -!True_emp plainly_pure. Qed.

  Lemma plainly_and_sep_l P Q : P Q ⊣⊢ P Q.
  Proof.
    apply (anti_symm (⊢));
      eauto using plainly_and_sep_l_1, sep_and with typeclass_instances.
  Qed.
  Lemma plainly_and_sep_r P Q : P Q ⊣⊢ P Q.
  Proof. by rewrite !(comm _ P) plainly_and_sep_l. Qed.

  Lemma plainly_impl_wand P Q : (P Q) ⊣⊢ (P -∗ Q).
  Proof.
    apply (anti_symm (⊢)); auto using plainly_impl_wand_2.
    apply plainly_intro', wand_intro_l.
    by rewrite -plainly_and_sep_r plainly_elim impl_elim_r.
  Qed.

  Lemma impl_wand_plainly P Q : ( P Q) ⊣⊢ ( P -∗ Q).
  Proof.
    apply (anti_symm (⊢)). by rewrite -impl_wand_1. by rewrite impl_wand_plainly_2.
  Qed.
End plainly_affine_bi.

Global Instance plainly_if_ne p : NonExpansive (@plainly_if PROP _ p).
Proof. solve_proper. Qed.
Global Instance plainly_if_proper p : Proper ((⊣⊢) ==> (⊣⊢)) (@plainly_if PROP _ p).
Proof. solve_proper. Qed.
Global Instance plainly_if_mono' p : Proper ((⊢) ==> (⊢)) (@plainly_if PROP _ p).
Proof. solve_proper. Qed.
Global Instance plainly_if_flip_mono' p :
  Proper (flip (⊢) ==> flip (⊢)) (@plainly_if PROP _ p).
Proof. solve_proper. Qed.

Lemma plainly_if_mono p P Q : (P Q) ■?p P ■?p Q.
Proof. by intros →. Qed.

Lemma plainly_if_pure p φ : ■?p φ ⊣⊢@{PROP} φ.
Proof. destruct p; simpl; auto using plainly_pure. Qed.
Lemma plainly_if_and p P Q : ■?p (P Q) ⊣⊢ ■?p P ■?p Q.
Proof. destruct p; simpl; auto using plainly_and. Qed.
Lemma plainly_if_or_2 p P Q : ■?p P ■?p Q ■?p (P Q).
Proof. destruct p; simpl; auto using plainly_or_2. Qed.
Lemma plainly_if_or `{!BiPlainlyExist PROP} p P Q : ■?p (P Q) ⊣⊢ ■?p P ■?p Q.
Proof. destruct p; simpl; auto using plainly_or. Qed.
Lemma plainly_if_exist_2 {A} p (Ψ : A PROP) : ( a, ■?p (Ψ a)) ■?p ( a, Ψ a).
Proof. destruct p; simpl; auto using plainly_exist_2. Qed.
Lemma plainly_if_exist `{!BiPlainlyExist PROP} {A} p (Ψ : A PROP) :
  ■?p ( a, Ψ a) ⊣⊢ a, ■?p (Ψ a).
Proof. destruct p; simpl; auto using plainly_exist. Qed.
Lemma plainly_if_sep_2 `{!BiPositive PROP} p P Q : ■?p P ■?p Q ■?p (P Q).
Proof. destruct p; simpl; auto using plainly_sep_2. Qed.

Lemma plainly_if_idemp p P : ■?p ■?p P ⊣⊢ ■?p P.
Proof. destruct p; simpl; auto using plainly_idemp. Qed.

Global Instance Plain_proper : Proper ((≡) ==> iff) (@Plain PROP _).
Proof. solve_proper. Qed.

Lemma plain_plainly_2 P `{!Plain P} : P P.
Proof. done. Qed.
Lemma plain_plainly P `{!Plain P, !Absorbing P} : P ⊣⊢ P.
Proof. apply (anti_symm _), plain_plainly_2, _. by apply plainly_elim. Qed.
Lemma plainly_intro P Q `{!Plain P} : (P Q) P Q.
Proof. by intros <-. Qed.

Global Instance plainly_absorbing P : Absorbing ( P).
Proof. by rewrite /Absorbing /bi_absorbingly comm plainly_absorb. Qed.
Global Instance plainly_if_absorbing P p :
  Absorbing P Absorbing (plainly_if p P).
Proof. intros; destruct p; simpl; apply _. Qed.

Lemma plain_persistent P : Plain P Persistent P.
Proof. intros. by rewrite /Persistent -plainly_elim_persistently. Qed.

Lemma impl_persistent P Q :
  Absorbing P Plain P Persistent Q Persistent (P Q).
Proof.
  intros. by rewrite /Persistent {2}(plain P) -persistently_impl_plainly
                     -(persistent Q) (plainly_into_absorbingly P) absorbing.
Qed.

Global Instance plainly_persistent P : Persistent ( P).
Proof. by rewrite /Persistent persistently_elim_plainly. Qed.

Global Instance wand_persistent P Q :
  Plain P Persistent Q Absorbing Q Persistent (P -∗ Q).
Proof.
  intros. rewrite /Persistent {2}(plain P). trans (<pers> ( P Q))%I.
  - rewrite -persistently_impl_plainly impl_wand_affinely_plainly -(persistent Q).
    by rewrite affinely_plainly_elim.
  - apply persistently_mono, wand_intro_l. by rewrite sep_and impl_elim_r.
Qed.

Global Instance limit_preserving_Plain {A:ofeT} `{Cofe A} (Φ : A PROP) :
  NonExpansive Φ LimitPreserving (λ x, Plain (Φ x)).
Proof. intros. apply limit_preserving_entails; solve_proper. Qed.

Global Instance plainly_sep_weak_homomorphism `{!BiPositive PROP, !BiAffine PROP} :
  WeakMonoidHomomorphism bi_sep bi_sep (≡) (@plainly PROP _).
Proof. split; try apply _. apply plainly_sep. Qed.
Global Instance plainly_sep_entails_weak_homomorphism :
  WeakMonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@plainly PROP _).
Proof. split; try apply _. intros P Q; by rewrite plainly_sep_2. Qed.
Global Instance plainly_sep_entails_homomorphism `{!BiAffine PROP} :
  MonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@plainly PROP _).
Proof. split. apply _. simpl. rewrite plainly_emp. done. Qed.

Global Instance plainly_sep_homomorphism `{BiAffine PROP} :
  MonoidHomomorphism bi_sep bi_sep (≡) (@plainly PROP _).
Proof. split. apply _. apply plainly_emp. Qed.
Global Instance plainly_and_homomorphism :
  MonoidHomomorphism bi_and bi_and (≡) (@plainly PROP _).
Proof. split; [split|]; try apply _. apply plainly_and. apply plainly_pure. Qed.
Global Instance plainly_or_homomorphism `{!BiPlainlyExist PROP} :
  MonoidHomomorphism bi_or bi_or (≡) (@plainly PROP _).
Proof. split; [split|]; try apply _. apply plainly_or. apply plainly_pure. Qed.

Lemma big_sepL_plainly `{!BiAffine PROP} {A} (Φ : nat A PROP) l :
   ([∗ list] kx l, Φ k x) ⊣⊢ [∗ list] kx l, (Φ k x).
Proof. apply (big_opL_commute _). Qed.
Lemma big_andL_plainly {A} (Φ : nat A PROP) l :
   ([∧ list] kx l, Φ k x) ⊣⊢ [∧ list] kx l, (Φ k x).
Proof. apply (big_opL_commute _). Qed.
Lemma big_orL_plainly `{!BiPlainlyExist PROP} {A} (Φ : nat A PROP) l :
   ([∨ list] kx l, Φ k x) ⊣⊢ [∨ list] kx l, (Φ k x).
Proof. apply (big_opL_commute _). Qed.

Lemma big_sepL2_plainly `{!BiAffine PROP} {A B} (Φ : nat A B PROP) l1 l2 :
   ([∗ list] ky1;y2 l1;l2, Φ k y1 y2)
  ⊣⊢ [∗ list] ky1;y2 l1;l2, (Φ k y1 y2).
Proof. by rewrite !big_sepL2_alt plainly_and plainly_pure big_sepL_plainly. Qed.

Lemma big_sepM_plainly `{BiAffine PROP, Countable K} {A} (Φ : K A PROP) m :
   ([∗ map] kx m, Φ k x) ⊣⊢ [∗ map] kx m, (Φ k x).
Proof. apply (big_opM_commute _). Qed.

Lemma big_sepM2_plainly `{BiAffine PROP, Countable K} {A B} (Φ : K A B PROP) m1 m2 :
   ([∗ map] kx1;x2 m1;m2, Φ k x1 x2) ⊣⊢ [∗ map] kx1;x2 m1;m2, (Φ k x1 x2).
Proof. by rewrite big_sepM2_eq /big_sepM2_def plainly_and plainly_pure big_sepM_plainly. Qed.

Lemma big_sepS_plainly `{BiAffine PROP, Countable A} (Φ : A PROP) X :
   ([∗ set] y X, Φ y) ⊣⊢ [∗ set] y X, (Φ y).
Proof. apply (big_opS_commute _). Qed.

Lemma big_sepMS_plainly `{BiAffine PROP, Countable A} (Φ : A PROP) X :
   ([∗ mset] y X, Φ y) ⊣⊢ [∗ mset] y X, (Φ y).
Proof. apply (big_opMS_commute _). Qed.

Global Instance pure_plain φ : Plain (PROP:=PROP) φ.
Proof. by rewrite /Plain plainly_pure. Qed.
Global Instance emp_plain : Plain (PROP:=PROP) emp.
Proof. apply plainly_emp_intro. Qed.
Global Instance and_plain P Q : Plain P Plain Q Plain (P Q).
Proof. intros. by rewrite /Plain plainly_and -!plain. Qed.
Global Instance or_plain P Q : Plain P Plain Q Plain (P Q).
Proof. intros. by rewrite /Plain -plainly_or_2 -!plain. Qed.
Global Instance forall_plain {A} (Ψ : A PROP) :
  ( x, Plain (Ψ x)) Plain ( x, Ψ x).
Proof.
  intros. rewrite /Plain plainly_forall. apply forall_monox. by rewrite -plain.
Qed.
Global Instance exist_plain {A} (Ψ : A PROP) :
  ( x, Plain (Ψ x)) Plain ( x, Ψ x).
Proof.
  intros. rewrite /Plain -plainly_exist_2. apply exist_monox. by rewrite -plain.
Qed.

Global Instance impl_plain P Q : Absorbing P Plain P Plain Q Plain (P Q).
Proof.
  intros. by rewrite /Plain {2}(plain P) -plainly_impl_plainly -(plain Q)
                     (plainly_into_absorbingly P) absorbing.
Qed.
Global Instance wand_plain P Q :
  Plain P Plain Q Absorbing Q Plain (P -∗ Q).
Proof.
  intros. rewrite /Plain {2}(plain P). trans ( (■ P Q))%I.
  - rewrite -plainly_impl_plainly impl_wand_affinely_plainly -(plain Q).
    by rewrite affinely_plainly_elim.
  - apply plainly_mono, wand_intro_l. by rewrite sep_and impl_elim_r.
Qed.
Global Instance sep_plain P Q : Plain P Plain Q Plain (P Q).
Proof. intros. by rewrite /Plain -plainly_sep_2 -!plain. Qed.

Global Instance plainly_plain P : Plain ( P).
Proof. by rewrite /Plain plainly_idemp. Qed.
Global Instance persistently_plain P : Plain P Plain (<pers> P).
Proof.
  rewrite /PlainHP. rewrite {1}HP plainly_persistently_elim persistently_elim_plainly //.
Qed.
Global Instance affinely_plain P : Plain P Plain (<affine> P).
Proof. rewrite /bi_affinely. apply _. Qed.
Global Instance intuitionistically_plain P : Plain P Plain ( P).
Proof. rewrite /bi_intuitionistically. apply _. Qed.
Global Instance absorbingly_plain P : Plain P Plain (<absorb> P).
Proof. rewrite /bi_absorbingly. apply _. Qed.
Global Instance from_option_plain {A} P (Ψ : A PROP) (mx : option A) :
  ( x, Plain (Ψ x)) Plain P Plain (from_option Ψ P mx).
Proof. destruct mx; apply _. Qed.

Global Instance big_sepL_nil_plain `{!BiAffine PROP} {A} (Φ : nat A PROP) :
  Plain ([∗ list] kx [], Φ k x).
Proof. simpl; apply _. Qed.
Global Instance big_sepL_plain `{!BiAffine PROP} {A} (Φ : nat A PROP) l :
  ( k x, Plain (Φ k x)) Plain ([∗ list] kx l, Φ k x).
Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
Global Instance big_andL_nil_plain {A} (Φ : nat A PROP) :
  Plain ([∧ list] kx [], Φ k x).
Proof. simpl; apply _. Qed.
Global Instance big_andL_plain {A} (Φ : nat A PROP) l :
  ( k x, Plain (Φ k x)) Plain ([∧ list] kx l, Φ k x).
Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
Global Instance big_orL_nil_plain {A} (Φ : nat A PROP) :
  Plain ([∨ list] kx [], Φ k x).
Proof. simpl; apply _. Qed.
Global Instance big_orL_plain {A} (Φ : nat A PROP) l :
  ( k x, Plain (Φ k x)) Plain ([∨ list] kx l, Φ k x).
Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.

Global Instance big_sepL2_nil_plain `{!BiAffine PROP} {A B} (Φ : nat A B PROP) :
  Plain ([∗ list] ky1;y2 []; [], Φ k y1 y2).
Proof. simpl; apply _. Qed.
Global Instance big_sepL2_plain `{!BiAffine PROP} {A B} (Φ : nat A B PROP) l1 l2 :
  ( k x1 x2, Plain (Φ k x1 x2))
  Plain ([∗ list] ky1;y2 l1;l2, Φ k y1 y2).
Proof. rewrite big_sepL2_alt. apply _. Qed.

Global Instance big_sepM_empty_plain `{BiAffine PROP, Countable K} {A} (Φ : K A PROP) :
  Plain ([∗ map] kx , Φ k x).
Proof. rewrite big_opM_eq /big_opM_def map_to_list_empty. apply _. Qed.
Global Instance big_sepM_plain `{BiAffine PROP, Countable K} {A} (Φ : K A PROP) m :
  ( k x, Plain (Φ k x)) Plain ([∗ map] kx m, Φ k x).
Proof. rewrite big_opM_eq. intros. apply (big_sepL_plain _ _)=> _ [??]; apply _. Qed.

Global Instance big_sepM2_empty_plain `{BiAffine PROP, Countable K}
    {A B} (Φ : K A B PROP) :
  Plain ([∗ map] kx1;x2 ;, Φ k x1 x2).
Proof. rewrite big_sepM2_eq /big_sepM2_def map_zip_with_empty. apply _. Qed.
Global Instance big_sepM2_plain `{BiAffine PROP, Countable K}
    {A B} (Φ : K A B PROP) m1 m2 :
  ( k x1 x2, Plain (Φ k x1 x2))
  Plain ([∗ map] kx1;x2 m1;m2, Φ k x1 x2).
Proof. intros. rewrite big_sepM2_eq. apply _. Qed.

Global Instance big_sepS_empty_plain `{BiAffine PROP, Countable A} (Φ : A PROP) :
  Plain ([∗ set] x , Φ x).
Proof. rewrite big_opS_eq /big_opS_def elements_empty. apply _. Qed.
Global Instance big_sepS_plain `{BiAffine PROP, Countable A} (Φ : A PROP) X :
  ( x, Plain (Φ x)) Plain ([∗ set] x X, Φ x).
Proof. rewrite big_opS_eq. apply _. Qed.

Global Instance big_sepMS_empty_plain `{BiAffine PROP, Countable A} (Φ : A PROP) :
  Plain ([∗ mset] x , Φ x).
Proof. rewrite big_opMS_eq /big_opMS_def gmultiset_elements_empty. apply _. Qed.
Global Instance big_sepMS_plain `{BiAffine PROP, Countable A} (Φ : A PROP) X :
  ( x, Plain (Φ x)) Plain ([∗ mset] x X, Φ x).
Proof. rewrite big_opMS_eq. apply _. Qed.

Section internal_eq.
  Context `{!BiInternalEq PROP}.

  Lemma plainly_internal_eq {A:ofeT} (a b : A) : (a b) ⊣⊢@{PROP} a b.
  Proof.
    apply (anti_symm (⊢)).
    { by rewrite plainly_elim. }
    apply (internal_eq_rewrite' a b (λ b, (a b))%I); [solve_proper|done|].
    rewrite -(internal_eq_refl True%I a) plainly_pure; auto.
  Qed.

  Global Instance internal_eq_plain {A : ofeT} (a b : A) :
    Plain (PROP:=PROP) (a b).
  Proof. by intros; rewrite /Plain plainly_internal_eq. Qed.
End internal_eq.

Section prop_ext.
  Context `{!BiInternalEq PROP, !BiPropExt PROP}.

  Lemma prop_ext P Q : P Q ⊣⊢ (P ∗-∗ Q).
  Proof.
    apply (anti_symm (⊢)); last exact: prop_ext_2.
    apply (internal_eq_rewrite' P Q (λ Q, (P ∗-∗ Q))%I);
      [ solve_proper | done | ].
    rewrite (plainly_emp_intro (P Q)%I).
    apply plainly_mono, wand_iff_refl.
  Qed.

  Lemma plainly_alt P : P ⊣⊢ <affine> P emp.
  Proof.
    rewrite -plainly_affinely_elim. apply (anti_symm (⊢)).
    - rewrite prop_ext. apply plainly_mono, and_intro; apply wand_intro_l.
      + by rewrite affinely_elim_emp left_id.
      + by rewrite left_id.
    - rewrite internal_eq_sym (internal_eq_rewrite _ _ plainly).
      by rewrite -plainly_True_emp plainly_pure True_impl.
  Qed.

  Lemma plainly_alt_absorbing P `{!Absorbing P} : P ⊣⊢ P True.
  Proof.
    apply (anti_symm (⊢)).
    - rewrite prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
    - rewrite internal_eq_sym (internal_eq_rewrite _ _ plainly).
      by rewrite plainly_pure True_impl.
  Qed.

  Lemma plainly_True_alt P : (True -∗ P) ⊣⊢ P True.
  Proof.
    apply (anti_symm (⊢)).
    - rewrite prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
      by rewrite wand_elim_r.
    - rewrite internal_eq_sym (internal_eq_rewrite _ _
        (λ Q, (True -∗ Q))%I ltac:(shelve)); last solve_proper.
      by rewrite -entails_wand // -(plainly_emp_intro True%I) True_impl.
  Qed.
End prop_ext.

Lemma later_plainly P : P ⊣⊢ P.
Proof. apply (anti_symm _); auto using later_plainly_1, later_plainly_2. Qed.
Lemma laterN_plainly n P : ▷^n P ⊣⊢ ▷^n P.
Proof. induction n as [|n IH]; simpl; auto. by rewrite IH later_plainly. Qed.

Lemma later_plainly_if p P : ■?p P ⊣⊢ ■?p P.
Proof. destruct p; simpl; auto using later_plainly. Qed.
Lemma laterN_plainly_if n p P : ▷^n ■?p P ⊣⊢ ■?p (▷^n P).
Proof. destruct p; simpl; auto using laterN_plainly. Qed.

Lemma except_0_plainly_1 P : P P.
Proof. by rewrite /bi_except_0 -plainly_or_2 -later_plainly plainly_pure. Qed.
Lemma except_0_plainly `{!BiPlainlyExist PROP} P : P ⊣⊢ P.
Proof. by rewrite /bi_except_0 plainly_or -later_plainly plainly_pure. Qed.

Global Instance later_plain P : Plain P Plain ( P).
Proof. intros. by rewrite /Plain -later_plainly {1}(plain P). Qed.
Global Instance laterN_plain n P : Plain P Plain (▷^n P).
Proof. induction n; apply _. Qed.
Global Instance except_0_plain P : Plain P Plain ( P).
Proof. rewrite /bi_except_0; apply _. Qed.

Global Instance plainly_timeless P `{!BiPlainlyExist PROP} :
  Timeless P Timeless ( P).
Proof.
  intros. rewrite /Timeless /bi_except_0 later_plainly_1.
  by rewrite (timeless P) /bi_except_0 plainly_or {1}plainly_elim.
Qed.
End plainly_derived.

Hint Immediate plain_persistent : typeclass_instances.

Hint Extern 4 (Persistent _) ⇒ notypeclasses refine (impl_persistent _ _ _ _ _) : typeclass_instances.