upred.v

From iris.algebra Require Export cmra updates.
From iris.bi Require Export derived_connectives updates plainly.
From stdpp Require Import finite.
Set Default Proof Using "Type".
Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|].
Local Hint Extern 1 (_ ≼ _) => etrans; [|eassumption].
Local Hint Extern 10 (_ ≤ _) => omega.

(** The basic definition of the uPred type, its metric and functor laws.
    You probably do not want to import this file. Instead, import
    base_logic.base_logic; that will also give you all the primitive
    and many derived laws for the logic. *)

(* A good way of understanding this definition of the uPred OFE is to
   consider the OFE uPred0 of monotonous SProp predicates. That is,
   uPred0 is the OFE of non-expansive functions from M to SProp that
   are monotonous with respect to CMRA inclusion. This notion of
   monotonicity has to be stated in the SProp logic. Together with the
   usual closedness property of SProp, this gives exactly uPred_mono.

   Then, we quotient uPred0 *in the sProp logic* with respect to
   equivalence on valid elements of M. That is, we quotient with
   respect to the following *sProp* equivalence relation:
     P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x))       (1)
   When seen from the ambiant logic, obtaining this quotient requires
   definig both a custom Equiv and Dist.


   It is worth noting that this equivalence relation admits canonical
   representatives. More precisely, one can show that every
   equivalence class contains exactly one element P0 such that:
     ∀ x, (✓ x → P0(x)) → P0(x)                 (2)
   (Again, this assertion has to be understood in sProp). Intuitively,
   this says that P0 trivially holds whenever the resource is invalid.
   Starting from any element P, one can find this canonical
   representative by choosing:
     P0(x) := ✓ x → P(x)                        (3)

   Hence, as an alternative definition of uPred, we could use the set
   of canonical representatives (i.e., the subtype of monotonous
   sProp predicates that verify (2)). This alternative definition would
   save us from using a quotient. However, the definitions of the various
   connectives would get more complicated, because we have to make sure
   they all verify (2), which sometimes requires some adjustments. We
   would moreover need to prove one more property for every logical
   connective.
 *)

Record uPred (M : ucmraT) : Type := IProp {
  uPred_holds :> nat → M → Prop;

  uPred_mono n1 n2 x1 x2 :
    uPred_holds n1 x1 → x1 ≼{n1} x2 → n2 ≤ n1 → uPred_holds n2 x2
}.
Arguments uPred_holds {_} _ _ _ : simpl never.
Add Printing Constructor uPred.
Instance: Params (@uPred_holds) 3.

Bind Scope bi_scope with uPred.
Arguments uPred_holds {_} _%I _ _.

Section cofe.
  Context {M : ucmraT}.

  Inductive uPred_equiv' (P Q : uPred M) : Prop :=
    { uPred_in_equiv : ∀ n x, ✓{n} x → P n x ↔ Q n x }.
  Instance uPred_equiv : Equiv (uPred M) := uPred_equiv'.
  Inductive uPred_dist' (n : nat) (P Q : uPred M) : Prop :=
    { uPred_in_dist : ∀ n' x, n' ≤ n → ✓{n'} x → P n' x ↔ Q n' x }.
  Instance uPred_dist : Dist (uPred M) := uPred_dist'.
  Definition uPred_ofe_mixin : OfeMixin (uPred M).
  Proof.
    split.
    - intros P Q; split.
      + by intros HPQ n; split=> i x ??; apply HPQ.
      + intros HPQ; split=> n x ?; apply HPQ with n; auto.
    - intros n; split.
      + by intros P; split=> x i.
      + by intros P Q HPQ; split=> x i ??; symmetry; apply HPQ.
      + intros P Q Q' HP HQ; split=> i x ??.
        by trans (Q i x);[apply HP|apply HQ].
    - intros n P Q HPQ; split=> i x ??; apply HPQ; auto.
  Qed.
  Canonical Structure uPredC : ofeT := OfeT (uPred M) uPred_ofe_mixin.

  Program Definition uPred_compl : Compl uPredC := λ c,
    {| uPred_holds n x := ∀ n', n' ≤ n → ✓{n'}x → c n' n' x |}.
  Next Obligation.
    move=> /= c n1 n2 x1 x2 HP Hx12 Hn12 n3 Hn23 Hv. eapply uPred_mono.
    eapply HP, cmra_validN_includedN, cmra_includedN_le=>//; lia.
    eapply cmra_includedN_le=>//; lia. done.
  Qed.
  Global Program Instance uPred_cofe : Cofe uPredC := {| compl := uPred_compl |}.
  Next Obligation.
    intros n c; split=>i x Hin Hv.
    etrans; [|by symmetry; apply (chain_cauchy c i n)]. split=>H; [by apply H|].
    repeat intro. apply (chain_cauchy c n' i)=>//. by eapply uPred_mono.
  Qed.
End cofe.
Arguments uPredC : clear implicits.

Instance uPred_ne {M} (P : uPred M) n : Proper (dist n ==> iff) (P n).
Proof.
  intros x1 x2 Hx; split=> ?; eapply uPred_mono; eauto; by rewrite Hx.
Qed.
Instance uPred_proper {M} (P : uPred M) n : Proper ((≡) ==> iff) (P n).
Proof. by intros x1 x2 Hx; apply uPred_ne, equiv_dist. Qed.

Lemma uPred_holds_ne {M} (P Q : uPred M) n1 n2 x :
  P ≡{n2}≡ Q → n2 ≤ n1 → ✓{n2} x → Q n1 x → P n2 x.
Proof.
  intros [Hne] ???. eapply Hne; try done. eauto using uPred_mono, cmra_validN_le.
Qed.

(* Equivalence to the definition of uPred in the appendix. *)
Lemma uPred_alt {M : ucmraT} (P: nat → M → Prop) :
  (∀ n1 n2 x1 x2, P n1 x1 → x1 ≼{n1} x2 → n2 ≤ n1 → P n2 x2) ↔
  ( (∀ x n1 n2, n2 ≤ n1 → P n1 x → P n2 x) (* Pointwise down-closed *)
  ∧ (∀ n x1 x2, x1 ≡{n}≡ x2 → ∀ m, m ≤ n → P m x1 ↔ P m x2) (* Non-expansive *)
  ∧ (∀ n x1 x2, x1 ≼{n} x2 → ∀ m, m ≤ n → P m x1 → P m x2) (* Monotonicity *)
  ).
Proof.
  (* Provide this lemma to eauto. *)
  assert (∀ n1 n2 (x1 x2 : M), n2 ≤ n1 → x1 ≡{n1}≡ x2 → x1 ≼{n2} x2).
  { intros ????? H. eapply cmra_includedN_le; last done. by rewrite H. }
  (* Now go ahead. *)
  split.
  - intros Hupred. repeat split; eauto using cmra_includedN_le.
  - intros (Hdown & _ & Hmono) **. eapply Hmono; [done..|]. eapply Hdown; done.
Qed.

(** functor *)
Program Definition uPred_map {M1 M2 : ucmraT} (f : M2 -n> M1)
  `{!CmraMorphism f} (P : uPred M1) :
  uPred M2 := {| uPred_holds n x := P n (f x) |}.
Next Obligation. naive_solver eauto using uPred_mono, cmra_morphism_monotoneN. Qed.

Instance uPred_map_ne {M1 M2 : ucmraT} (f : M2 -n> M1)
  `{!CmraMorphism f} n : Proper (dist n ==> dist n) (uPred_map f).
Proof.
  intros x1 x2 Hx; split=> n' y ??.
  split; apply Hx; auto using cmra_morphism_validN.
Qed.
Lemma uPred_map_id {M : ucmraT} (P : uPred M): uPred_map cid P ≡ P.
Proof. by split=> n x ?. Qed.
Lemma uPred_map_compose {M1 M2 M3 : ucmraT} (f : M1 -n> M2) (g : M2 -n> M3)
    `{!CmraMorphism f, !CmraMorphism g} (P : uPred M3):
  uPred_map (g ◎ f) P ≡ uPred_map f (uPred_map g P).
Proof. by split=> n x Hx. Qed.
Lemma uPred_map_ext {M1 M2 : ucmraT} (f g : M1 -n> M2)
      `{!CmraMorphism f} `{!CmraMorphism g}:
  (∀ x, f x ≡ g x) → ∀ x, uPred_map f x ≡ uPred_map g x.
Proof. intros Hf P; split=> n x Hx /=; by rewrite /uPred_holds /= Hf. Qed.
Definition uPredC_map {M1 M2 : ucmraT} (f : M2 -n> M1) `{!CmraMorphism f} :
  uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2).
Lemma uPredC_map_ne {M1 M2 : ucmraT} (f g : M2 -n> M1)
    `{!CmraMorphism f, !CmraMorphism g} n :
  f ≡{n}≡ g → uPredC_map f ≡{n}≡ uPredC_map g.
Proof.
  by intros Hfg P; split=> n' y ??;
    rewrite /uPred_holds /= (dist_le _ _ _ _(Hfg y)); last lia.
Qed.

Program Definition uPredCF (F : urFunctor) : cFunctor := {|
  cFunctor_car A B := uPredC (urFunctor_car F B A);
  cFunctor_map A1 A2 B1 B2 fg := uPredC_map (urFunctor_map F (fg.2, fg.1))
|}.
Next Obligation.
  intros F A1 A2 B1 B2 n P Q HPQ.
  apply uPredC_map_ne, urFunctor_ne; split; by apply HPQ.
Qed.
Next Obligation.
  intros F A B P; simpl. rewrite -{2}(uPred_map_id P).
  apply uPred_map_ext=>y. by rewrite urFunctor_id.
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' P; simpl. rewrite -uPred_map_compose.
  apply uPred_map_ext=>y; apply urFunctor_compose.
Qed.

Instance uPredCF_contractive F :
  urFunctorContractive F → cFunctorContractive (uPredCF F).
Proof.
  intros ? A1 A2 B1 B2 n P Q HPQ. apply uPredC_map_ne, urFunctor_contractive.
  destruct n; split; by apply HPQ.
Qed.

(** logical entailement *)
Inductive uPred_entails {M} (P Q : uPred M) : Prop :=
  { uPred_in_entails : ∀ n x, ✓{n} x → P n x → Q n x }.
Hint Resolve uPred_mono : uPred_def.

(** logical connectives *)
Program Definition uPred_pure_def {M} (φ : Prop) : uPred M :=
  {| uPred_holds n x := φ |}.
Solve Obligations with done.
Definition uPred_pure_aux : seal (@uPred_pure_def). by eexists. Qed.
Definition uPred_pure {M} := uPred_pure_aux.(unseal) M.
Definition uPred_pure_eq :
  @uPred_pure = @uPred_pure_def := uPred_pure_aux.(seal_eq).

Definition uPred_emp {M} : uPred M := uPred_pure True.

Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := P n x ∧ Q n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
Definition uPred_and_aux : seal (@uPred_and_def). by eexists. Qed.
Definition uPred_and {M} := uPred_and_aux.(unseal) M.
Definition uPred_and_eq: @uPred_and = @uPred_and_def := uPred_and_aux.(seal_eq).

Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := P n x ∨ Q n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
Definition uPred_or_aux : seal (@uPred_or_def). by eexists. Qed.
Definition uPred_or {M} := uPred_or_aux.(unseal) M.
Definition uPred_or_eq: @uPred_or = @uPred_or_def := uPred_or_aux.(seal_eq).

Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := ∀ n' x',
       x ≼ x' → n' ≤ n → ✓{n'} x' → P n' x' → Q n' x' |}.
Next Obligation.
  intros M P Q n1 n1' x1 x1' HPQ [x2 Hx1'] Hn1 n2 x3 [x4 Hx3] ?; simpl in *.
  rewrite Hx3 (dist_le _ _ _ _ Hx1'); auto. intros ??.
  eapply HPQ; auto. exists (x2 ⋅ x4); by rewrite assoc.
Qed.
Definition uPred_impl_aux : seal (@uPred_impl_def). by eexists. Qed.
Definition uPred_impl {M} := uPred_impl_aux.(unseal) M.
Definition uPred_impl_eq :
  @uPred_impl = @uPred_impl_def := uPred_impl_aux.(seal_eq).

Program Definition uPred_forall_def {M A} (Ψ : A → uPred M) : uPred M :=
  {| uPred_holds n x := ∀ a, Ψ a n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
Definition uPred_forall_aux : seal (@uPred_forall_def). by eexists. Qed.
Definition uPred_forall {M A} := uPred_forall_aux.(unseal) M A.
Definition uPred_forall_eq :
  @uPred_forall = @uPred_forall_def := uPred_forall_aux.(seal_eq).

Program Definition uPred_exist_def {M A} (Ψ : A → uPred M) : uPred M :=
  {| uPred_holds n x := ∃ a, Ψ a n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
Definition uPred_exist_aux : seal (@uPred_exist_def). by eexists. Qed.
Definition uPred_exist {M A} := uPred_exist_aux.(unseal) M A.
Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := uPred_exist_aux.(seal_eq).

Program Definition uPred_internal_eq_def {M} {A : ofeT} (a1 a2 : A) : uPred M :=
  {| uPred_holds n x := a1 ≡{n}≡ a2 |}.
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
Definition uPred_internal_eq_aux : seal (@uPred_internal_eq_def). by eexists. Qed.
Definition uPred_internal_eq {M A} := uPred_internal_eq_aux.(unseal) M A.
Definition uPred_internal_eq_eq:
  @uPred_internal_eq = @uPred_internal_eq_def := uPred_internal_eq_aux.(seal_eq).

Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}.
Next Obligation.
  intros M P Q n1 n2 x y (x1&x2&Hx&?&?) [z Hy] Hn.
  exists x1, (x2 ⋅ z); split_and?; eauto using uPred_mono, cmra_includedN_l.
  eapply dist_le, Hn. by rewrite Hy Hx assoc.
Qed.
Definition uPred_sep_aux : seal (@uPred_sep_def). by eexists. Qed.
Definition uPred_sep {M} := uPred_sep_aux.(unseal) M.
Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := uPred_sep_aux.(seal_eq).

Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := ∀ n' x',
       n' ≤ n → ✓{n'} (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}.
Next Obligation.
  intros M P Q n1 n1' x1 x1' HPQ ? Hn n3 x3 ???; simpl in *.
  eapply uPred_mono with n3 (x1 ⋅ x3);
    eauto using cmra_validN_includedN, cmra_monoN_r, cmra_includedN_le.
Qed.
Definition uPred_wand_aux : seal (@uPred_wand_def). by eexists. Qed.
Definition uPred_wand {M} := uPred_wand_aux.(unseal) M.
Definition uPred_wand_eq :
  @uPred_wand = @uPred_wand_def := uPred_wand_aux.(seal_eq).

(* Equivalently, this could be `∀ y, P n y`.  That's closer to the intuition
   of "embedding the step-indexed logic in Iris", but the two are equivalent
   because Iris is afine.  The following is easier to work with. *)
Program Definition uPred_plainly_def {M} : Plainly (uPred M) := λ P,
  {| uPred_holds n x := P n ε |}.
Solve Obligations with naive_solver eauto using uPred_mono, ucmra_unit_validN.
Definition uPred_plainly_aux : seal (@uPred_plainly_def). by eexists. Qed.
Definition uPred_plainly {M} := uPred_plainly_aux.(unseal) M.
Definition uPred_plainly_eq :
  @uPred_plainly = @uPred_plainly_def := uPred_plainly_aux.(seal_eq).

Program Definition uPred_persistently_def {M} (P : uPred M) : uPred M :=
  {| uPred_holds n x := P n (core x) |}.
Next Obligation.
  intros M; naive_solver eauto using uPred_mono, @cmra_core_monoN.
Qed.
Definition uPred_persistently_aux : seal (@uPred_persistently_def). by eexists. Qed.
Definition uPred_persistently {M} := uPred_persistently_aux.(unseal) M.
Definition uPred_persistently_eq :
  @uPred_persistently = @uPred_persistently_def := uPred_persistently_aux.(seal_eq).

Program Definition uPred_later_def {M} (P : uPred M) : uPred M :=
  {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
Next Obligation.
  intros M P [|n1] [|n2] x1 x2; eauto using uPred_mono, cmra_includedN_S with lia.
Qed.
Definition uPred_later_aux : seal (@uPred_later_def). by eexists. Qed.
Definition uPred_later {M} := uPred_later_aux.(unseal) M.
Definition uPred_later_eq :
  @uPred_later = @uPred_later_def := uPred_later_aux.(seal_eq).

Program Definition uPred_ownM_def {M : ucmraT} (a : M) : uPred M :=
  {| uPred_holds n x := a ≼{n} x |}.
Next Obligation.
  intros M a n1 n2 x1 x [a' Hx1] [x2 Hx] Hn. eapply cmra_includedN_le=>//.
  exists (a' ⋅ x2). by rewrite Hx(assoc op) Hx1.
Qed.
Definition uPred_ownM_aux : seal (@uPred_ownM_def). by eexists. Qed.
Definition uPred_ownM {M} := uPred_ownM_aux.(unseal) M.
Definition uPred_ownM_eq :
  @uPred_ownM = @uPred_ownM_def := uPred_ownM_aux.(seal_eq).

Program Definition uPred_cmra_valid_def {M} {A : cmraT} (a : A) : uPred M :=
  {| uPred_holds n x := ✓{n} a |}.
Solve Obligations with naive_solver eauto 2 using cmra_validN_le.
Definition uPred_cmra_valid_aux : seal (@uPred_cmra_valid_def). by eexists. Qed.
Definition uPred_cmra_valid {M A} := uPred_cmra_valid_aux.(unseal) M A.
Definition uPred_cmra_valid_eq :
  @uPred_cmra_valid = @uPred_cmra_valid_def := uPred_cmra_valid_aux.(seal_eq).

Program Definition uPred_bupd_def {M} (Q : uPred M) : uPred M :=
  {| uPred_holds n x := ∀ k yf,
      k ≤ n → ✓{k} (x ⋅ yf) → ∃ x', ✓{k} (x' ⋅ yf) ∧ Q k x' |}.
Next Obligation.
  intros M Q n1 n2 x1 x2 HQ [x3 Hx] Hn k yf Hk.
  rewrite (dist_le _ _ _ _ Hx); last lia. intros Hxy.
  destruct (HQ k (x3 ⋅ yf)) as (x'&?&?); [auto|by rewrite assoc|].
  exists (x' ⋅ x3); split; first by rewrite -assoc.
  eauto using uPred_mono, cmra_includedN_l.
Qed.
Definition uPred_bupd_aux {M} : seal (@uPred_bupd_def M). by eexists. Qed.
Definition uPred_bupd {M} : BUpd (uPred M) := uPred_bupd_aux.(unseal).
Definition uPred_bupd_eq {M} :
  @bupd _ uPred_bupd = @uPred_bupd_def M := uPred_bupd_aux.(seal_eq).

Module uPred_unseal.
Definition unseal_eqs :=
  (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
  uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq,
  uPred_plainly_eq, uPred_persistently_eq, uPred_later_eq, uPred_ownM_eq,
  uPred_cmra_valid_eq, @uPred_bupd_eq).
Ltac unseal := (* Coq unfold is used to circumvent bug #5699 in rewrite /foo *)
  unfold bi_emp; simpl; unfold sbi_emp; simpl;
  unfold uPred_emp, bi_pure, bi_and, bi_or, bi_impl, bi_forall, bi_exist,
  bi_sep, bi_wand, bi_persistently, sbi_internal_eq, sbi_later; simpl;
  unfold sbi_emp, sbi_pure, sbi_and, sbi_or, sbi_impl, sbi_forall, sbi_exist,
  sbi_internal_eq, sbi_sep, sbi_wand, sbi_persistently; simpl;
  unfold plainly, bi_plainly_plainly; simpl;
  rewrite !unseal_eqs /=.
End uPred_unseal.
Import uPred_unseal.

Local Arguments uPred_holds {_} !_ _ _ /.

Lemma uPred_bi_mixin (M : ucmraT) :
  BiMixin
    uPred_entails uPred_emp uPred_pure uPred_and uPred_or uPred_impl
    (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand
    uPred_persistently.
Proof.
  split.
  - (* PreOrder uPred_entails *)
    split.
    + by intros P; split=> x i.
    + by intros P Q Q' HP HQ; split=> x i ??; apply HQ, HP.
  - (* (P ⊣⊢ Q) ↔ (P ⊢ Q) ∧ (Q ⊢ P) *)
    intros P Q. split.
    + intros HPQ; split; split=> x i; apply HPQ.
    + intros [HPQ HQP]; split=> x n; by split; [apply HPQ|apply HQP].
  - (* Proper (iff ==> dist n) (@uPred_pure M) *)
    intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ.
  - (* NonExpansive2 uPred_and *)
    intros n P P' HP Q Q' HQ; unseal; split=> x n' ??.
    split; (intros [??]; split; [by apply HP|by apply HQ]).
  - (* NonExpansive2 uPred_or *)
    intros n P P' HP Q Q' HQ; split=> x n' ??.
    unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]).
  - (* NonExpansive2 uPred_impl *)
    intros n P P' HP Q Q' HQ; split=> x n' ??.
    unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto.
  - (* Proper (pointwise_relation A (dist n) ==> dist n) uPred_forall *)
    by intros A n Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
  - (* Proper (pointwise_relation A (dist n) ==> dist n) uPred_exist *)
    intros A n Ψ1 Ψ2 HΨ.
    unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ.
  - (* NonExpansive2 uPred_sep *)
    intros n P P' HP Q Q' HQ; split=> n' x ??.
    unseal; split; intros (x1&x2&?&?&?); ofe_subst x;
      exists x1, x2; split_and!; try (apply HP || apply HQ);
      eauto using cmra_validN_op_l, cmra_validN_op_r.
  - (* NonExpansive2 uPred_wand *)
    intros n P P' HP Q Q' HQ; split=> n' x ??.
    unseal; split; intros HPQ x' n'' ???;
      apply HQ, HPQ, HP; eauto using cmra_validN_op_r.
  - (* NonExpansive uPred_persistently *)
    intros n P1 P2 HP.
    unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN.
  - (* φ → P ⊢ ⌜φ⌝ *)
    intros P φ ?. unseal; by split.
  - (* (φ → True ⊢ P) → ⌜φ⌝ ⊢ P *)
    intros φ P. unseal=> HP; split=> n x ??. by apply HP.
  - (* (∀ x : A, ⌜φ x⌝) ⊢ ⌜∀ x : A, φ x⌝ *)
    by unseal.
  - (* P ∧ Q ⊢ P *)
    intros P Q. unseal; by split=> n x ? [??].
  - (* P ∧ Q ⊢ Q *)
    intros P Q. unseal; by split=> n x ? [??].
  - (* (P ⊢ Q) → (P ⊢ R) → P ⊢ Q ∧ R *)
    intros P Q R HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR].
  - (* P ⊢ P ∨ Q *)
    intros P Q. unseal; split=> n x ??; left; auto.
  - (* Q ⊢ P ∨ Q *)
    intros P Q. unseal; split=> n x ??; right; auto.
  - (* (P ⊢ R) → (Q ⊢ R) → P ∨ Q ⊢ R *)
    intros P Q R HP HQ. unseal; split=> n x ? [?|?]. by apply HP. by apply HQ.
  - (* (P ∧ Q ⊢ R) → P ⊢ Q → R. *)
    intros P Q R. unseal => HQ; split=> n x ?? n' x' ????. apply HQ;
      naive_solver eauto using uPred_mono, cmra_included_includedN.
  - (* (P ⊢ Q → R) → P ∧ Q ⊢ R *)
    intros P Q R. unseal=> HP; split=> n x ? [??]. apply HP with n x; auto.
  - (* (∀ a, P ⊢ Ψ a) → P ⊢ ∀ a, Ψ a *)
    intros A P Ψ. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ.
  - (* (∀ a, Ψ a) ⊢ Ψ a *)
    intros A Ψ a. unseal; split=> n x ? HP; apply HP.
  - (* Ψ a ⊢ ∃ a, Ψ a *)
    intros A Ψ a. unseal; split=> n x ??; by exists a.
  - (* (∀ a, Ψ a ⊢ Q) → (∃ a, Ψ a) ⊢ Q *)
    intros A Ψ Q. unseal; intros HΨ; split=> n x ? [a ?]; by apply HΨ with a.
  - (* (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q' *)
    intros P P' Q Q' HQ HQ'; unseal.
    split; intros n' x ? (x1&x2&?&?&?); exists x1,x2; ofe_subst x;
      eauto 7 using cmra_validN_op_l, cmra_validN_op_r, uPred_in_entails.
  - (* P ⊢ emp ∗ P *)
    intros P. rewrite /uPred_emp. unseal; split=> n x ?? /=.
    exists (core x), x. by rewrite cmra_core_l.
  - (* emp ∗ P ⊢ P *)
    intros P. unseal; split; intros n x ? (x1&x2&?&_&?); ofe_subst;
      eauto using uPred_mono, cmra_includedN_r.
  - (* P ∗ Q ⊢ Q ∗ P *)
    intros P Q. unseal; split; intros n x ? (x1&x2&?&?&?).
    exists x2, x1; by rewrite (comm op).
  - (* (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R) *)
    intros P Q R. unseal; split; intros n x ? (x1&x2&Hx&(y1&y2&Hy&?&?)&?).
    exists y1, (y2 ⋅ x2); split_and?; auto.
    + by rewrite (assoc op) -Hy -Hx.
    + by exists y2, x2.
  - (* (P ∗ Q ⊢ R) → P ⊢ Q -∗ R *)
    intros P Q R. unseal=> HPQR; split=> n x ?? n' x' ???; apply HPQR; auto.
    exists x, x'; split_and?; auto.
    eapply uPred_mono; eauto using cmra_validN_op_l.
  - (* (P ⊢ Q -∗ R) → P ∗ Q ⊢ R *)
    intros P Q R. unseal=> HPQR. split; intros n x ? (?&?&?&?&?). ofe_subst.
    eapply HPQR; eauto using cmra_validN_op_l.
  - (* (P ⊢ Q) → <pers> P ⊢ <pers> Q *)
    intros P QR HP. unseal; split=> n x ? /=. by apply HP, cmra_core_validN.
  - (* <pers> P ⊢ <pers> <pers> P *)
    intros P. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp.
  - (* P ⊢ <pers> emp (ADMISSIBLE) *)
    by unseal.
  - (* (∀ a, <pers> (Ψ a)) ⊢ <pers> (∀ a, Ψ a) *)
    by unseal.
  - (* <pers> (∃ a, Ψ a) ⊢ ∃ a, <pers> (Ψ a) *)
    by unseal.
  - (* <pers> P ∗ Q ⊢ <pers> P (ADMISSIBLE) *)
    intros P Q. move: (uPred_persistently P)=> P'.
    unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst;
      eauto using uPred_mono, cmra_includedN_l.
  - (* <pers> P ∧ Q ⊢ P ∗ Q *)
    intros P Q. unseal; split=> n x ? [??]; simpl in *.
    exists (core x), x; rewrite ?cmra_core_l; auto.
Qed.

Lemma uPred_sbi_mixin (M : ucmraT) : SbiMixin
  uPred_entails uPred_pure uPred_or uPred_impl
  (@uPred_forall M) (@uPred_exist M) uPred_sep
  uPred_persistently (@uPred_internal_eq M) uPred_later.
Proof.
  split.
  - (* Contractive sbi_later *)
    unseal; intros [|n] P Q HPQ; split=> -[|n'] x ?? //=; try omega.
    apply HPQ; eauto using cmra_validN_S.
  - (* NonExpansive2 (@uPred_internal_eq M A) *)
    intros A n x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *.
    + by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto.
    + by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto.
  - (* P ⊢ a ≡ a *)
    intros A P a. unseal; by split=> n x ?? /=.
  - (* a ≡ b ⊢ Ψ a → Ψ b *)
    intros A a b Ψ Hnonexp.
    unseal; split=> n x ? Hab n' x' ??? HΨ. eapply Hnonexp with n a; auto.
  - (* (∀ x, f x ≡ g x) ⊢ f ≡ g *)
    by unseal.
  - (* `x ≡ `y ⊢ x ≡ y *)
    by unseal.
  - (* Discrete a → a ≡ b ⊣⊢ ⌜a ≡ b⌝ *)
    intros A a b ?. unseal; split=> n x ?; by apply (discrete_iff n).
  - (* Next x ≡ Next y ⊢ ▷ (x ≡ y) *)
    by unseal.
  - (* ▷ (x ≡ y) ⊢ Next x ≡ Next y *)
    by unseal.
  - (* (P ⊢ Q) → ▷ P ⊢ ▷ Q *)
    intros P Q.
    unseal=> HP; split=>-[|n] x ??; [done|apply HP; eauto using cmra_validN_S].
  - (* P ⊢ ▷ P *)
    intros P. unseal; split=> -[|n] /= x ? HP; first done.
    apply uPred_mono with (S n) x; eauto using cmra_validN_S.
  - (* (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a *)
    intros A Φ. unseal; by split=> -[|n] x.
  - (* (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a) *)
    intros A Φ. unseal; split=> -[|[|n]] x /=; eauto.
  - (* ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q *)
    intros P Q. unseal; split=> -[|n] x ? /=.
    { by exists x, (core x); rewrite cmra_core_r. }
    intros (x1&x2&Hx&?&?); destruct (cmra_extend n x x1 x2)
      as (y1&y2&Hx'&Hy1&Hy2); eauto using cmra_validN_S; simpl in *.
    exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1 Hy2].
  - (* ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q) *)
    intros P Q. unseal; split=> -[|n] x ? /=; [done|intros (x1&x2&Hx&?&?)].
    exists x1, x2; eauto using dist_S.
  - (* ▷ <pers> P ⊢ <pers> ▷ P *)
    by unseal.
  - (* <pers> ▷ P ⊢ ▷ <pers> P *)
    by unseal.
  - (* ▷ P ⊢ ▷ False ∨ (▷ False → P) *)
    intros P. unseal; split=> -[|n] x ? /= HP; [by left|right].
    intros [|n'] x' ????; [|done].
    eauto using uPred_mono, cmra_included_includedN.
Qed.

Canonical Structure uPredI (M : ucmraT) : bi :=
  {| bi_ofe_mixin := ofe_mixin_of (uPred M); bi_bi_mixin := uPred_bi_mixin M |}.
Canonical Structure uPredSI (M : ucmraT) : sbi :=
  {| sbi_ofe_mixin := ofe_mixin_of (uPred M);
     sbi_bi_mixin := uPred_bi_mixin M; sbi_sbi_mixin := uPred_sbi_mixin M |}.

Coercion uPred_valid {M} : uPred M → Prop := bi_emp_valid.

(* Latest notation *)
Notation "✓ x" := (uPred_cmra_valid x) (at level 20) : bi_scope.

Lemma uPred_plainly_mixin M : BiPlainlyMixin (uPredSI M) uPred_plainly.
Proof.
  split.
  - (* NonExpansive uPred_plainly *)
    intros n P1 P2 HP.
    unseal; split=> n' x; split; apply HP; eauto using @ucmra_unit_validN.
  - (* (P ⊢ Q) → ■ P ⊢ ■ Q *)
    intros P QR HP. unseal; split=> n x ? /=. by apply HP, ucmra_unit_validN.
  - (* ■ P ⊢ <pers> P *)
    unseal; split; simpl; eauto using uPred_mono, @ucmra_unit_leastN.
  - (* ■ P ⊢ ■ ■ P *)
    unseal; split=> n x ?? //.
  - (* (∀ a, ■ (Ψ a)) ⊢ ■ (∀ a, Ψ a) *)
    by unseal.
  - (* (■ P → <pers> Q) ⊢ <pers> (■ P → Q) *)
    unseal; split=> /= n x ? HPQ n' x' ????.
    eapply uPred_mono with n' (core x)=>//; [|by apply cmra_included_includedN].
    apply (HPQ n' x); eauto using cmra_validN_le.
  - (* (■ P → ■ Q) ⊢ ■ (■ P → Q) *)
    unseal; split=> /= n x ? HPQ n' x' ????.
    eapply uPred_mono with n' ε=>//; [|by apply cmra_included_includedN].
    apply (HPQ n' x); eauto using cmra_validN_le.
  - (* P ⊢ ■ emp (ADMISSIBLE) *)
    by unseal.
  - (* ■ P ∗ Q ⊢ ■ P *)
    intros P Q. move: (uPred_persistently P)=> P'.
    unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst;
      eauto using uPred_mono, cmra_includedN_l.
  - (* ■ ((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q *)
    unseal; split=> n x ? /= HPQ. split=> n' x' ??.
    move: HPQ=> [] /(_ n' x'); rewrite !left_id=> ?.
    move=> /(_ n' x'); rewrite !left_id=> ?. naive_solver.
  - (* ▷ ■ P ⊢ ■ ▷ P *)
    by unseal.
  - (* ■ ▷ P ⊢ ▷ ■ P *)
    by unseal.
Qed.
Instance uPred_plainlyC M : BiPlainly (uPredSI M) :=
  {| bi_plainly_mixin := uPred_plainly_mixin M |}.

Lemma uPred_bupd_mixin M : BiBUpdMixin (uPredI M) uPred_bupd.
Proof.
  split.
  - intros n P Q HPQ.
    unseal; split=> n' x; split; intros HP k yf ??;
    destruct (HP k yf) as (x'&?&?); auto;
    exists x'; split; auto; apply HPQ; eauto using cmra_validN_op_l.
  - unseal. split=> n x ? HP k yf ?; exists x; split; first done.
    apply uPred_mono with n x; eauto using cmra_validN_op_l.
  - unseal. intros HPQ; split=> n x ? HP k yf ??.
    destruct (HP k yf) as (x'&?&?); eauto.
    exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l.
  - unseal; split; naive_solver.
  - unseal. split; intros n x ? (x1&x2&Hx&HP&?) k yf ??.
    destruct (HP k (x2 ⋅ yf)) as (x'&?&?); eauto.
    { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. }
    exists (x' ⋅ x2); split; first by rewrite -assoc.
    exists x', x2. eauto using uPred_mono, cmra_validN_op_l, cmra_validN_op_r.
Qed.
Instance uPred_bi_bupd M : BiBUpd (uPredI M) := {| bi_bupd_mixin := uPred_bupd_mixin M |}.

Instance uPred_bi_bupd_plainly M : BiBUpdPlainly (uPredSI M).
Proof.
  rewrite /BiBUpdPlainly. unseal; split => n x Hnx /= Hng.
  destruct (Hng n ε) as [? [_ Hng']]; try rewrite right_id; auto.
  eapply uPred_mono; eauto using ucmra_unit_leastN.
Qed.

Module uPred.
Include uPred_unseal.
Section uPred.
Context {M : ucmraT}.
Implicit Types φ : Prop.
Implicit Types P Q : uPred M.
Implicit Types A : Type.
Arguments uPred_holds {_} !_ _ _ /.
Hint Immediate uPred_in_entails.

Global Instance ownM_ne : NonExpansive (@uPred_ownM M).
Proof.
  intros n a b Ha.
  unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia.
Qed.
Global Instance ownM_proper: Proper ((≡) ==> (⊣⊢)) (@uPred_ownM M) := ne_proper _.

Global Instance cmra_valid_ne {A : cmraT} :
  NonExpansive (@uPred_cmra_valid M A).
Proof.
  intros n a b Ha; unseal; split=> n' x ? /=.
  by rewrite (dist_le _ _ _ _ Ha); last lia.
Qed.
Global Instance cmra_valid_proper {A : cmraT} :
  Proper ((≡) ==> (⊣⊢)) (@uPred_cmra_valid M A) := ne_proper _.

(** BI instances *)

Global Instance uPred_affine : BiAffine (uPredI M) | 0.
Proof. intros P. rewrite /Affine. by apply bi.pure_intro. Qed.

Global Instance uPred_plainly_exist_1 : BiPlainlyExist (uPredSI M).
Proof. unfold BiPlainlyExist. by unseal. Qed.

(** Limits *)
Lemma entails_lim (cP cQ : chain (uPredC M)) :
  (∀ n, cP n ⊢ cQ n) → compl cP ⊢ compl cQ.
Proof.
  intros Hlim; split=> n m ? HP.
  eapply uPred_holds_ne, Hlim, HP; eauto using conv_compl.
Qed.

(* Own *)
Lemma ownM_op (a1 a2 : M) :
  uPred_ownM (a1 ⋅ a2) ⊣⊢ uPred_ownM a1 ∗ uPred_ownM a2.
Proof.
  rewrite /bi_sep /=; unseal. split=> n x ?; split.
  - intros [z ?]; exists a1, (a2 ⋅ z); split; [by rewrite (assoc op)|].
    split. by exists (core a1); rewrite cmra_core_r. by exists z.
  - intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1 ⋅ z2).
    by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1)
      -(assoc op _ a2) (comm op z1) -Hy1 -Hy2.
Qed.
Lemma persistently_ownM_core (a : M) : uPred_ownM a ⊢ <pers> uPred_ownM (core a).
Proof.
  rewrite /bi_persistently /=. unseal.
  split=> n x Hx /=. by apply cmra_core_monoN.
Qed.
Lemma ownM_unit : bi_emp_valid (uPred_ownM (ε:M)).
Proof. unseal; split=> n x ??; by  exists x; rewrite left_id. Qed.
Lemma later_ownM (a : M) : ▷ uPred_ownM a ⊢ ∃ b, uPred_ownM b ∧ ▷ (a ≡ b).
Proof.
  rewrite /bi_and /sbi_later /bi_exist /sbi_internal_eq /=; unseal.
  split=> -[|n] x /= ? Hax; first by eauto using ucmra_unit_leastN.
  destruct Hax as [y ?].
  destruct (cmra_extend n x a y) as (a'&y'&Hx&?&?); auto using cmra_validN_S.
  exists a'. rewrite Hx. eauto using cmra_includedN_l.
Qed.

(* Valid *)
Lemma discrete_valid {A : cmraT} `{!CmraDiscrete A} (a : A) :
  ✓ a ⊣⊢ (⌜✓ a⌝ : uPred M).
Proof. unseal. split=> n x _. by rewrite /= -cmra_discrete_valid_iff. Qed.
Lemma ownM_valid (a : M) : uPred_ownM a ⊢ ✓ a.
Proof.
  unseal; split=> n x Hv [a' ?]; ofe_subst; eauto using cmra_validN_op_l.
Qed.
Lemma cmra_valid_intro {A : cmraT} (a : A) :
  ✓ a → bi_emp_valid (PROP:=uPredI M) (✓ a).
Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed.
Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a → ✓ a ⊢ (False : uPred M).
Proof.
  intros Ha. unseal. split=> n x ??; apply Ha, cmra_validN_le with n; auto.
Qed.
Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ ■ (✓ a : uPred M).
Proof. by unseal. Qed.
Lemma cmra_valid_weaken {A : cmraT} (a b : A) : ✓ (a ⋅ b) ⊢ (✓ a : uPred M).
Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed.

Lemma prod_validI {A B : cmraT} (x : A * B) : ✓ x ⊣⊢ (✓ x.1 ∧ ✓ x.2 : uPred M).
Proof. by unseal. Qed.
Lemma option_validI {A : cmraT} (mx : option A) :
  ✓ mx ⊣⊢ match mx with Some x => ✓ x | None => True : uPred M end.
Proof. unseal. by destruct mx. Qed.

Lemma ofe_fun_validI `{B : A → ucmraT} (g : ofe_fun B) :
  (✓ g : uPred M) ⊣⊢ ∀ i, ✓ g i.
Proof. by uPred.unseal. Qed.

Lemma bupd_ownM_updateP x (Φ : M → Prop) :
  x ~~>: Φ → uPred_ownM x ==∗ ∃ y, ⌜Φ y⌝ ∧ uPred_ownM y.
Proof.
  intros Hup. unseal. split=> n x2 ? [x3 Hx] k yf ??.
  destruct (Hup k (Some (x3 ⋅ yf))) as (y&?&?); simpl in *.
  { rewrite /= assoc -(dist_le _ _ _ _ Hx); auto. }
  exists (y ⋅ x3); split; first by rewrite -assoc.
  exists y; eauto using cmra_includedN_l.
Qed.
End uPred.
End uPred.