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Require Import iris.cmra.
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Local Hint Extern 1 (_  _) => etransitivity; [eassumption|].
Local Hint Extern 1 (_  _) => etransitivity; [|eassumption].
Local Hint Extern 10 (_  _) => omega.

Structure iProp (A : cmraT) : Type := IProp {
  iprop_holds :> nat  A -> Prop;
  iprop_ne x1 x2 n : iprop_holds n x1  x1 ={n}= x2  iprop_holds n x2;
  iprop_weaken x1 x2 n1 n2 :
    x1  x2  n2  n1  validN n2 x2  iprop_holds n1 x1  iprop_holds n2 x2
}.
Add Printing Constructor iProp.
Instance: Params (@iprop_holds) 3.

Instance iprop_equiv (A : cmraT) : Equiv (iProp A) := λ P Q,  x n,
  validN n x  P n x  Q n x.
Instance iprop_dist (A : cmraT) : Dist (iProp A) := λ n P Q,  x n',
  n' < n  validN n' x  P n' x  Q n' x.
Program Instance iprop_compl (A : cmraT) : Compl (iProp A) := λ c,
  {| iprop_holds n x := c (S n) n x |}.
Next Obligation. by intros A c x y n ??; simpl in *; apply iprop_ne with x. Qed.
Next Obligation.
  intros A c x1 x2 n1 n2 ????; simpl in *.
  apply (chain_cauchy c (S n2) (S n1)); eauto using iprop_weaken, cmra_valid_le.
Qed.
Instance iprop_cofe (A : cmraT) : Cofe (iProp A).
Proof.
  split.
  * intros P Q; split; [by intros HPQ n x i ??; apply HPQ|].
    intros HPQ x n ?; apply HPQ with (S n); auto.
  * intros n; split.
    + by intros P x i.
    + by intros P Q HPQ x i ??; symmetry; apply HPQ.
    + by intros P Q Q' HP HQ x i ??; transitivity (Q i x); [apply HP|apply HQ].
  * intros n P Q HPQ x i ??; apply HPQ; auto.
  * intros P Q x i ??; lia.
  * intros c n x i ??; apply (chain_cauchy c (S i) n); auto.
Qed.
Instance iprop_holds_ne {A} (P : iProp A) n : Proper (dist n ==> iff) (P n).
Proof. intros x1 x2 Hx; split; eauto using iprop_ne. Qed.
Instance iprop_holds_proper {A} (P : iProp A) n : Proper (() ==> iff) (P n).
Proof. by intros x1 x2 Hx; apply iprop_holds_ne, equiv_dist. Qed.
Definition iPropC (A : cmraT) : cofeT := CofeT (iProp A).

(** functor *)
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Program Definition iprop_map {A B : cmraT} (f : B  A)
  `{! n, Proper (dist n ==> dist n) f, !CMRAPreserving f}
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  (P : iProp A) : iProp B := {| iprop_holds n x := P n (f x) |}.
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Next Obligation. by intros A B f ?? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed.
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Next Obligation.
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  by intros A B f ?? P y1 y2 n i ???; simpl; apply iprop_weaken; auto;
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    apply validN_preserving || apply included_preserving.
Qed.
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Instance iprop_map_ne {A B : cmraT} (f : B  A)
  `{! n, Proper (dist n ==> dist n) f, !CMRAPreserving f} :
  Proper (dist n ==> dist n) (iprop_map f).
Proof.
  by intros n x1 x2 Hx y n'; split; apply Hx; try apply validN_preserving.
Qed.
Definition ipropC_map {A B : cmraT} (f : B -n> A) `{!CMRAPreserving f} :
  iPropC A -n> iPropC B := CofeMor (iprop_map f : iPropC A  iPropC B).
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(** logical entailement *)
Instance iprop_entails {A} : SubsetEq (iProp A) := λ P Q,  x n,
  validN n x  P n x  Q n x.

(** logical connectives *)
Program Definition iprop_const {A} (P : Prop) : iProp A :=
  {| iprop_holds n x := P |}.
Solve Obligations with done.

Program Definition iprop_and {A} (P Q : iProp A) : iProp A :=
  {| iprop_holds n x := P n x  Q n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_or {A} (P Q : iProp A) : iProp A :=
  {| iprop_holds n x := P n x  Q n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_impl {A} (P Q : iProp A) : iProp A :=
  {| iprop_holds n x :=  x' n',
       x  x'  n'  n  validN n' x'  P n' x'  Q n' x' |}.
Next Obligation.
  intros A P Q x1' x1 n1 HPQ Hx1 x2 n2 ????.
  destruct (cmra_included_dist_l x1 x2 x1' n1) as (x2'&?&Hx2); auto.
  assert (x2' ={n2}= x2) as Hx2' by (by apply dist_le with n1).
  assert (validN n2 x2') by (by rewrite Hx2'); rewrite <-Hx2'.
  by apply HPQ, iprop_weaken with x2' n2, iprop_ne with x2.
Qed.
Next Obligation. naive_solver eauto 2 with lia. Qed.

Program Definition iprop_forall {A B} (P : A  iProp B) : iProp B :=
  {| iprop_holds n x :=  a, P a n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_exist {A B} (P : A  iProp B) : iProp B :=
  {| iprop_holds n x :=  a, P a n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.

Program Definition iprop_eq {A} {B : cofeT} (b1 b2 : B) : iProp A :=
  {| iprop_holds n x := b1 ={n}= b2 |}.
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=B)).

Program Definition iprop_sep {A} (P Q : iProp A) : iProp A :=
  {| iprop_holds n x :=  x1 x2, x ={n}= x1  x2  P n x1  Q n x2 |}.
Next Obligation.
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  by intros A P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy.
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Qed.
Next Obligation.
  intros A P Q x y n1 n2 Hxy ?? (x1&x2&Hx&?&?).
  assert ( x2', y ={n2}= x1  x2'  x2  x2') as (x2'&Hy&?).
  { rewrite ra_included_spec in Hxy; destruct Hxy as [z Hy].
    exists (x2  z); split; eauto using ra_included_l.
    apply dist_le with n1; auto. by rewrite (associative op), <-Hx, Hy. }
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  exists x1, x2'; split_ands; auto.
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  * apply iprop_weaken with x1 n1; auto.
    by apply cmra_valid_op_l with x2'; rewrite <-Hy.
  * apply iprop_weaken with x2 n1; auto.
    by apply cmra_valid_op_r with x1; rewrite <-Hy.
Qed.

Program Definition iprop_wand {A} (P Q : iProp A) : iProp A :=
  {| iprop_holds n x :=  x' n',
       n'  n  validN n' (x  x')  P n' x'  Q n' (x  x') |}.
Next Obligation.
  intros A P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *.
  rewrite <-(dist_le _ _ _ _ Hx) by done; apply HPQ; auto.
  by rewrite (dist_le _ _ _ n2 Hx).
Qed.
Next Obligation.
  intros A P Q x1 x2 n1 n2 ??? HPQ x3 n3 ???; simpl in *.
  apply iprop_weaken with (x1  x3) n3; auto using ra_preserving_r.
  apply HPQ; auto.
  apply cmra_valid_included with (x2  x3); auto using ra_preserving_r.
Qed.

Program Definition iprop_later {A} (P : iProp A) : iProp A :=
  {| iprop_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
Next Obligation. intros A P ?? [|n]; eauto using iprop_ne,(dist_le (A:=A)). Qed.
Next Obligation.
  intros A P x1 x2 [|n1] [|n2] ????; auto with lia.
  apply iprop_weaken with x1 n1; eauto using cmra_valid_S.
Qed.
Program Definition iprop_always {A} (P : iProp A) : iProp A :=
  {| iprop_holds n x := P n (unit x) |}.
Next Obligation. by intros A P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed.
Next Obligation.
  intros A P x1 x2 n1 n2 ????; eapply iprop_weaken with (unit x1) n1;
    auto using ra_unit_preserving, cmra_unit_valid.
Qed.

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Program Definition iprop_own {A : cmraT} (a : A) : iProp A :=
  {| iprop_holds n x :=  a', x ={n}= a  a' |}.
Next Obligation. by intros A a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed.
Next Obligation.
  intros A a x1 x n1 n2; rewrite ra_included_spec; intros [x2 Hx] ?? [a' Hx1].
  exists (a'  x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx.
Qed.
Program Definition iprop_valid {A : cmraT} (a : A) : iProp A :=
  {| iprop_holds n x := validN n a |}.
Solve Obligations with naive_solver eauto 2 using cmra_valid_le.

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Definition iprop_fixpoint {A} (P : iProp A  iProp A)
  `{!Contractive P} : iProp A := fixpoint P (iprop_const True).

Delimit Scope iprop_scope with I.
Bind Scope iprop_scope with iProp.
Arguments iprop_holds {_} _%I _ _.

Notation "'False'" := (iprop_const False) : iprop_scope.
Notation "'True'" := (iprop_const True) : iprop_scope.
Infix "∧" := iprop_and : iprop_scope.
Infix "∨" := iprop_or : iprop_scope.
Infix "→" := iprop_impl : iprop_scope.
Infix "★" := iprop_sep (at level 80, right associativity) : iprop_scope.
Infix "-★" := iprop_wand (at level 90) : iprop_scope.
Notation "∀ x .. y , P" :=
  (iprop_forall (λ x, .. (iprop_forall (λ y, P)) ..)) : iprop_scope.
Notation "∃ x .. y , P" :=
  (iprop_exist (λ x, .. (iprop_exist (λ y, P)) ..)) : iprop_scope.
Notation "▷ P" := (iprop_later P) (at level 20) : iprop_scope.
Notation "□ P" := (iprop_always P) (at level 20) : iprop_scope.

Section logic.
Context {A : cmraT}.
Implicit Types P Q : iProp A.

Global Instance iprop_preorder : PreOrder (() : relation (iProp A)).
Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
Lemma iprop_equiv_spec P Q : P  Q  P  Q  Q  P.
Proof.
  split.
  * intros HPQ; split; intros x i; apply HPQ.
  * by intros [HPQ HQP]; intros x i ?; split; [apply HPQ|apply HQP].
Qed.

(** Non-expansiveness *)
Global Instance iprop_const_proper : Proper (iff ==> ()) (@iprop_const A).
Proof. intros P Q HPQ ???; apply HPQ. Qed.
Global Instance iprop_and_ne n :
  Proper (dist n ==> dist n ==> dist n) (@iprop_and A).
Proof.
  intros P P' HP Q Q' HQ; split; intros [??]; split; by apply HP || by apply HQ.
Qed.
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Global Instance iprop_and_proper :
  Proper (() ==> () ==> ()) (@iprop_and A) := ne_proper_2 _.
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Global Instance iprop_or_ne n :
  Proper (dist n ==> dist n ==> dist n) (@iprop_or A).
Proof.
  intros P P' HP Q Q' HQ; split; intros [?|?];
    first [by left; apply HP | by right; apply HQ].
Qed.
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Global Instance iprop_or_proper :
  Proper (() ==> () ==> ()) (@iprop_or A) := ne_proper_2 _.
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Global Instance iprop_impl_ne n :
  Proper (dist n ==> dist n ==> dist n) (@iprop_impl A).
Proof.
  intros P P' HP Q Q' HQ; split; intros HPQ x' n'' ????; apply HQ,HPQ,HP; auto.
Qed.
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Global Instance iprop_impl_proper :
  Proper (() ==> () ==> ()) (@iprop_impl A) := ne_proper_2 _.
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Global Instance iprop_sep_ne n :
  Proper (dist n ==> dist n ==> dist n) (@iprop_sep A).
Proof.
  intros P P' HP Q Q' HQ x n' ? Hx'; split; intros (x1&x2&Hx&?&?);
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    exists x1, x2; rewrite  Hx in Hx'; split_ands; try apply HP; try apply HQ;
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    eauto using cmra_valid_op_l, cmra_valid_op_r.
Qed.
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Global Instance iprop_sep_proper :
  Proper (() ==> () ==> ()) (@iprop_sep A) := ne_proper_2 _.
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Global Instance iprop_wand_ne n :
  Proper (dist n ==> dist n ==> dist n) (@iprop_wand A).
Proof.
  intros P P' HP Q Q' HQ x n' ??; split; intros HPQ x' n'' ???;
    apply HQ, HPQ, HP; eauto using cmra_valid_op_r.
Qed.
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Global Instance iprop_wand_proper :
  Proper (() ==> () ==> ()) (@iprop_wand A) := ne_proper_2 _.
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Global Instance iprop_eq_ne {B : cofeT} n :
  Proper (dist n ==> dist n ==> dist n) (@iprop_eq A B).
Proof.
  intros x x' Hx y y' Hy z n'; split; intros; simpl in *.
  * by rewrite <-(dist_le _ _ _ _ Hx), <-(dist_le _ _ _ _ Hy) by auto.
  * by rewrite (dist_le _ _ _ _ Hx), (dist_le _ _ _ _ Hy) by auto.
Qed.
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Global Instance iprop_eq_proper {B : cofeT} :
  Proper (() ==> () ==> ()) (@iprop_eq A B) := ne_proper_2 _.
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Global Instance iprop_forall_ne {B : cofeT} :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_forall B A).
Proof. by intros n P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
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Global Instance iprop_forall_proper {B : cofeT} :
  Proper (pointwise_relation _ () ==> ()) (@iprop_forall B A).
Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
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Global Instance iprop_exists_ne {B : cofeT} :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_exist B A).
Proof.
  by intros n P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
Qed.
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Global Instance iprop_exist_proper {B : cofeT} :
  Proper (pointwise_relation _ () ==> ()) (@iprop_exist B A).
Proof.
  by intros P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
Qed.
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Global Instance iprop_later_contractive : Contractive (@iprop_later A).
Proof.
  intros n P Q HPQ x [|n'] ??; simpl; [done|].
  apply HPQ; eauto using cmra_valid_S.
Qed.
Global Instance iprop_later_proper :
  Proper (() ==> ()) (@iprop_later A) := ne_proper _.
Global Instance iprop_always_ne n: Proper (dist n ==> dist n) (@iprop_always A).
Proof. intros P1 P2 HP x n'; split; apply HP; eauto using cmra_unit_valid. Qed.
Global Instance iprop_always_proper :
  Proper (() ==> ()) (@iprop_always A) := ne_proper _.
Global Instance iprop_own_ne n : Proper (dist n ==> dist n) (@iprop_own A).
Proof.
  by intros a1 a2 Ha x n'; split; intros [a' ?]; exists a'; simpl; first
    [rewrite <-(dist_le _ _ _ _ Ha) by lia|rewrite (dist_le _ _ _ _ Ha) by lia].
Qed.
Global Instance iprop_own_proper :
  Proper (() ==> ()) (@iprop_own A) := ne_proper _.
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(** Introduction and elimination rules *)
Lemma iprop_True_intro P : P  True%I.
Proof. done. Qed.
Lemma iprop_False_elim P : False%I  P.
Proof. by intros x n ?. Qed.
Lemma iprop_and_elim_l P Q : (P  Q)%I  P.
Proof. by intros x n ? [??]. Qed.
Lemma iprop_and_elim_r P Q : (P  Q)%I  Q.
Proof. by intros x n ? [??]. Qed.
Lemma iprop_and_intro R P Q : R  P  R  Q  R  (P  Q)%I.
Proof. intros HP HQ x n ??; split. by apply HP. by apply HQ. Qed.
Lemma iprop_or_intro_l P Q : P  (P  Q)%I.
Proof. by left. Qed.
Lemma iprop_or_intro_r P Q : Q  (P  Q)%I.
Proof. by right. Qed.
Lemma iprop_or_elim R P Q : P  R  Q  R  (P  Q)%I  R.
Proof. intros HP HQ x n ? [?|?]. by apply HP. by apply HQ. Qed.
Lemma iprop_impl_intro P Q R : (R  P)%I  Q  R  (P  Q)%I.
Proof.
  intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using iprop_weaken.
Qed.
Lemma iprop_impl_elim P Q : ((P  Q)  P)%I  Q.
Proof. by intros x n ? [HQ HP]; apply HQ. Qed.
Lemma iprop_forall_intro P `(Q: B  iProp A): ( b, P  Q b)  P  ( b, Q b)%I.
Proof. by intros HPQ x n ?? b; apply HPQ. Qed.
Lemma iprop_forall_elim `(P : B  iProp A) b : ( b, P b)%I  P b.
Proof. intros x n ? HP; apply HP. Qed.
Lemma iprop_exist_intro `(P : B  iProp A) b : P b  ( b, P b)%I.
Proof. by intros x n ??; exists b. Qed.
Lemma iprop_exist_elim `(P : B  iProp A) Q : ( b, P b  Q)  ( b, P b)%I  Q.
Proof. by intros HPQ x n ? [b ?]; apply HPQ with b. Qed.

(* BI connectives *)
Lemma iprop_sep_elim_l P Q : (P  Q)%I  P.
Proof.
  intros x n Hvalid (x1&x2&Hx&?&?); rewrite Hx in Hvalid |- *.
  by apply iprop_weaken with x1 n; auto using ra_included_l.
Qed.
Global Instance iprop_sep_left_id : LeftId () True%I (@iprop_sep A).
Proof.
  intros P x n Hvalid; split.
  * intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
    apply iprop_weaken with x2 n; auto using ra_included_r.
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  * by intros ?; exists (unit x), x; rewrite ra_unit_l.
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Qed. 
Global Instance iprop_sep_commutative : Commutative () (@iprop_sep A).
Proof.
  by intros P Q x n ?; split;
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    intros (x1&x2&?&?&?); exists x2, x1; rewrite (commutative op).
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Qed.
Global Instance iprop_sep_associative : Associative () (@iprop_sep A).
Proof.
  intros P Q R x n ?; split.
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  * intros (x1&x2&Hx&?&y1&y2&Hy&?&?); exists (x1  y1), y2; split_ands; auto.
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    + by rewrite <-(associative op), <-Hy, <-Hx.
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    + by exists x1, y1.
  * intros (x1&x2&Hx&(y1&y2&Hy&?&?)&?); exists y1, (y2  x2); split_ands; auto.
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    + by rewrite (associative op), <-Hy, <-Hx.
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    + by exists y2, x2.
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Qed.
Lemma iprop_wand_intro P Q R : (R  P)%I  Q  R  (P - Q)%I.
Proof.
  intros HPQ x n ?? x' n' ???; apply HPQ; auto.
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  exists x, x'; split_ands; auto.
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  eapply iprop_weaken with x n; eauto using cmra_valid_op_l.
Qed.
Lemma iprop_wand_elim P Q : ((P - Q)  P)%I  Q.
Proof.
  by intros x n Hvalid (x1&x2&Hx&HPQ&?); rewrite Hx in Hvalid |- *; apply HPQ.
Qed.
Lemma iprop_sep_or P Q R : ((P  Q)  R)%I  ((P  R)  (Q  R))%I.
Proof.
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  split; [by intros (x1&x2&Hx&[?|?]&?); [left|right]; exists x1, x2|].
  intros [(x1&x2&Hx&?&?)|(x1&x2&Hx&?&?)]; exists x1, x2; split_ands;
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    first [by left | by right | done].
Qed.
Lemma iprop_sep_and P Q R : ((P  Q)  R)%I  ((P  R)  (Q  R))%I.
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Proof. by intros x n ? (x1&x2&Hx&[??]&?); split; exists x1, x2. Qed.
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Lemma iprop_sep_exist {B} (P : B  iProp A) Q :
  (( b, P b)  Q)%I  ( b, P b  Q)%I.
Proof.
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  split; [by intros (x1&x2&Hx&[b ?]&?); exists b, x1, x2|].
  intros [b (x1&x2&Hx&?&?)]; exists x1, x2; split_ands; by try exists b.
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Qed.
Lemma iprop_sep_forall `(P : B  iProp A) Q :
  (( b, P b)  Q)%I  ( b, P b  Q)%I.
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Proof. by intros x n ? (x1&x2&Hx&?&?); intros b; exists x1, x2. Qed.
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(* Later *)
Lemma iprop_later_weaken P : P  ( P)%I.
Proof.
  intros x [|n] ??; simpl in *; [done|].
  apply iprop_weaken with x (S n); auto using cmra_valid_S.
Qed.
Lemma iprop_lub P : ( P  P)%I  P.
Proof.
  intros x n ? HP; induction n as [|n IH]; [by apply HP|].
  apply HP, IH, iprop_weaken with x (S n); eauto using cmra_valid_S.
Qed.
Lemma iprop_later_impl P Q : ( (P  Q))%I  ( P   Q)%I.
Proof.
  intros x [|n] ? HPQ x' [|n'] ???; auto with lia.
  apply HPQ; auto using cmra_valid_S.
Qed.
Lemma iprop_later_and P Q : ( (P  Q))%I  ( P   Q)%I.
Proof. by intros x [|n]; split. Qed.
Lemma iprop_later_or P Q : ( (P  Q))%I  ( P   Q)%I.
Proof. intros x [|n]; simpl; tauto. Qed.
Lemma iprop_later_forall `(P : B  iProp A) : (  b, P b)%I  ( b,  P b)%I.
Proof. by intros x [|n]. Qed.
Lemma iprop_later_exist `(P : B  iProp A) : ( b,  P b)%I  (  b, P b)%I.
Proof. by intros x [|n]. Qed.
Lemma iprop_later_exist' `{Inhabited B} (P : B  iProp A) :
  (  b, P b)%I  ( b,  P b)%I.
Proof.
  intros x [|n]; split; try done.
  by destruct (_ : Inhabited B) as [b]; exists b.
Qed.
Lemma iprop_later_sep P Q : ( (P  Q))%I  ( P   Q)%I.
Proof.
  intros x n ?; split.
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  * destruct n as [|n]; simpl; [by exists x, x|intros (x1&x2&Hx&?&?)].
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    destruct (cmra_extend_op x x1 x2 n)
      as ([y1 y2]&Hx'&Hy1&Hy2); auto using cmra_valid_S; simpl in *.
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    exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1, Hy2].
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  * destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)].
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    exists x1, x2; eauto using (dist_S (A:=A)).
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Qed.

(* Always *)
Lemma iprop_always_necessity P : ( P)%I  P.
Proof.
  intros x n ??; apply iprop_weaken with (unit x) n;auto using ra_included_unit.
Qed.
Lemma iprop_always_intro P Q : ( P)%I  Q  ( P)%I  ( Q)%I.
Proof.
  intros HPQ x n ??; apply HPQ; simpl in *; auto using cmra_unit_valid.
  by rewrite ra_unit_idempotent.
Qed.
Lemma iprop_always_impl P Q : ( (P  Q))%I  (P  Q)%I.
Proof.
  intros x n ? HPQ x' n' ???.
  apply HPQ; auto using ra_unit_preserving, cmra_unit_valid.
Qed.
Lemma iprop_always_and P Q : ( (P  Q))%I  ( P   Q)%I.
Proof. done. Qed.
Lemma iprop_always_or P Q : ( (P  Q))%I  ( P   Q)%I.
Proof. done. Qed.
Lemma iprop_always_forall `(P : B  iProp A) : (  b, P b)%I  ( b,  P b)%I.
Proof. done. Qed.
Lemma iprop_always_exist `(P : B  iProp A) : (  b, P b)%I  ( b,  P b)%I.
Proof. done. Qed.
Lemma iprop_always_and_always_box P Q : ( P  Q)%I  ( P  Q)%I.
Proof.
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  intros x n ? [??]; exists (unit x), x; simpl in *.
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  by rewrite ra_unit_l, ra_unit_idempotent.
Qed.

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(* Own *)
Lemma iprop_own_op (a1 a2 : A) :
  iprop_own (a1  a2)  (iprop_own a1  iprop_own a2)%I.
Proof.
  intros x n ?; split.
  * intros [z ?]; exists a1, (a2  z); split; [by rewrite (associative op)|].
    split. by exists (unit a1); rewrite ra_unit_r. by exists z.
  * intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1  z2).
    rewrite (associative op), <-(commutative op z1), <-!(associative op), <-Hy2.
    by rewrite (associative op), (commutative op z1), <-Hy1.
Qed.
Lemma iprop_own_valid (a : A) : iprop_own a  iprop_valid a.
Proof.
  intros x n Hv [a' Hx]; simpl; rewrite Hx in Hv; eauto using cmra_valid_op_l.
Qed.

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(* Fix *)
Lemma iprop_fixpoint_unfold (P : iProp A  iProp A) `{!Contractive P} :
  iprop_fixpoint P  P (iprop_fixpoint P).
Proof. apply fixpoint_unfold. Qed.
End logic.