upred.v

From algebra Require Export cmra.
Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|].
Local Hint Extern 1 (_ ≼ _) => etrans; [|eassumption].
Local Hint Extern 10 (_ ≤ _) => omega.

Record uPred (M : cmraT) : Type := IProp {
  uPred_holds :> nat → M → Prop;
  uPred_ne n x1 x2 : uPred_holds n x1 → x1 ≡{n}≡ x2 → uPred_holds n x2;
  uPred_weaken  n1 n2 x1 x2 :
    uPred_holds n1 x1 → x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → uPred_holds n2 x2
}.
Arguments uPred_holds {_} _ _ _ : simpl never.
Add Printing Constructor uPred.
Instance: Params (@uPred_holds) 3.

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

Section cofe.
  Context {M : cmraT}.

  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'.
  Program Instance uPred_compl : Compl (uPred M) := λ c,
    {| uPred_holds n x := c n n x |}.
  Next Obligation. by intros c n x y ??; simpl in *; apply uPred_ne with x. Qed.
  Next Obligation.
    intros c n1 n2 x1 x2 ????; simpl in *.
    apply (chain_cauchy c n2 n1); eauto using uPred_weaken.
  Qed.
  Definition uPred_cofe_mixin : CofeMixin (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.
    - intros n c; split=>i x ??; symmetry; apply (chain_cauchy c i n); auto.
  Qed.
  Canonical Structure uPredC : cofeT := CofeT uPred_cofe_mixin.
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; eauto using uPred_ne. 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} (P1 P2 : uPred M) n x :
  P1 ≡{n}≡ P2 → ✓{n} x → P1 n x → P2 n x.
Proof. intros HP ?; apply HP; auto. Qed.
Lemma uPred_weaken' {M} (P : uPred M) n1 n2 x1 x2 :
  x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → P n1 x1 → P n2 x2.
Proof. eauto using uPred_weaken. Qed.

(** functor *)
Program Definition uPred_map {M1 M2 : cmraT} (f : M2 -n> M1)
  `{!CMRAMonotone f} (P : uPred M1) :
  uPred M2 := {| uPred_holds n x := P n (f x) |}.
Next Obligation. by intros M1 M2 f ? P y1 y2 n ? Hy; rewrite /= -Hy. Qed.
Next Obligation.
  naive_solver eauto using uPred_weaken, included_preserving, validN_preserving.
Qed.
Instance uPred_map_ne {M1 M2 : cmraT} (f : M2 -n> M1)
  `{!CMRAMonotone f} n : Proper (dist n ==> dist n) (uPred_map f).
Proof.
  intros x1 x2 Hx; split=> n' y ??.
  split; apply Hx; auto using validN_preserving.
Qed.
Lemma uPred_map_id {M : cmraT} (P : uPred M): uPred_map cid P ≡ P.
Proof. by split=> n x ?. Qed.
Lemma uPred_map_compose {M1 M2 M3 : cmraT} (f : M1 -n> M2) (g : M2 -n> M3)
    `{!CMRAMonotone f, !CMRAMonotone 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 : cmraT} (f g : M1 -n> M2)
      `{!CMRAMonotone f} `{!CMRAMonotone 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 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} :
  uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2).
Lemma uPredC_map_ne {M1 M2 : cmraT} (f g : M2 -n> M1)
    `{!CMRAMonotone f, !CMRAMonotone 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 : rFunctor) : cFunctor := {|
  cFunctor_car A B := uPredC (rFunctor_car F B A);
  cFunctor_map A1 A2 B1 B2 fg := uPredC_map (rFunctor_map F (fg.2, fg.1))
|}.
Next Obligation.
  intros F A1 A2 B1 B2 n P Q HPQ.
  apply uPredC_map_ne, rFunctor_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 rFunctor_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 rFunctor_compose.
Qed.

Instance uPredCF_contractive F :
  rFunctorContractive F → cFunctorContractive (uPredCF F).
Proof.
  intros ? A1 A2 B1 B2 n P Q HPQ.
  apply uPredC_map_ne, rFunctor_contractive=> i ?; 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 Extern 0 (uPred_entails _ _) => reflexivity.
Instance uPred_entails_rewrite_relation M : RewriteRelation (@uPred_entails M).

(** logical connectives *)
Program Definition uPred_const_def {M} (φ : Prop) : uPred M :=
  {| uPred_holds n x := φ |}.
Solve Obligations with done.
Definition uPred_const_aux : { x | x = @uPred_const_def }. by eexists. Qed.
Definition uPred_const {M} := proj1_sig uPred_const_aux M.
Definition uPred_const_eq :
  @uPred_const = @uPred_const_def := proj2_sig uPred_const_aux.

Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_const 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 using uPred_ne, uPred_weaken.
Definition uPred_and_aux : { x | x = @uPred_and_def }. by eexists. Qed.
Definition uPred_and {M} := proj1_sig uPred_and_aux M.
Definition uPred_and_eq: @uPred_and = @uPred_and_def := proj2_sig uPred_and_aux.

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 using uPred_ne, uPred_weaken.
Definition uPred_or_aux : { x | x = @uPred_or_def }. by eexists. Qed.
Definition uPred_or {M} := proj1_sig uPred_or_aux M.
Definition uPred_or_eq: @uPred_or = @uPred_or_def := proj2_sig uPred_or_aux.

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 x1' x1 HPQ Hx1 n2 x2 ????.
  destruct (cmra_included_dist_l n1 x1 x2 x1') as (x2'&?&Hx2); auto.
  assert (x2' ≡{n2}≡ x2) as Hx2' by (by apply dist_le with n1).
  assert (✓{n2} x2') by (by rewrite Hx2'); rewrite -Hx2'.
  eauto using uPred_weaken, uPred_ne.
Qed.
Next Obligation. intros M P Q [|n] x1 x2; auto with lia. Qed.
Definition uPred_impl_aux : { x | x = @uPred_impl_def }. by eexists. Qed.
Definition uPred_impl {M} := proj1_sig uPred_impl_aux M.
Definition uPred_impl_eq :
  @uPred_impl = @uPred_impl_def := proj2_sig uPred_impl_aux.

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 using uPred_ne, uPred_weaken.
Definition uPred_forall_aux : { x | x = @uPred_forall_def }. by eexists. Qed.
Definition uPred_forall {M A} := proj1_sig uPred_forall_aux M A.
Definition uPred_forall_eq :
  @uPred_forall = @uPred_forall_def := proj2_sig uPred_forall_aux.

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 using uPred_ne, uPred_weaken.
Definition uPred_exist_aux : { x | x = @uPred_exist_def }. by eexists. Qed.
Definition uPred_exist {M A} := proj1_sig uPred_exist_aux M A.
Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := proj2_sig uPred_exist_aux.

Program Definition uPred_eq_def {M} {A : cofeT} (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_eq_aux : { x | x = @uPred_eq_def }. by eexists. Qed.
Definition uPred_eq {M A} := proj1_sig uPred_eq_aux M A.
Definition uPred_eq_eq: @uPred_eq = @uPred_eq_def := proj2_sig uPred_eq_aux.

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.
  by intros M P Q n x y (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite -Hxy.
Qed.
Next Obligation.
  intros M P Q n1 n2 x y (x1&x2&Hx&?&?) Hxy ??.
  assert (∃ x2', y ≡{n2}≡ x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?).
  { destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using cmra_included_l.
    apply dist_le with n1; auto. by rewrite (assoc op) -Hx Hy. }
  clear Hxy; cofe_subst y; exists x1, x2'; split_and?; [done| |].
  - apply uPred_weaken with n1 x1; eauto using cmra_validN_op_l.
  - apply uPred_weaken with n1 x2; eauto using cmra_validN_op_r.
Qed.
Definition uPred_sep_aux : { x | x = @uPred_sep_def }. by eexists. Qed.
Definition uPred_sep {M} := proj1_sig uPred_sep_aux M.
Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := proj2_sig uPred_sep_aux.

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 x1 x2 HPQ Hx n2 x3 ???; simpl in *.
  rewrite -(dist_le _ _ _ _ Hx) //; apply HPQ; auto.
  by rewrite (dist_le _ _ _ _ Hx).
Qed.
Next Obligation.
  intros M P Q n1 n2 x1 x2 HPQ ??? n3 x3 ???; simpl in *.
  apply uPred_weaken with n3 (x1 ⋅ x3);
    eauto using cmra_validN_included, cmra_preserving_r.
Qed.
Definition uPred_wand_aux : { x | x = @uPred_wand_def }. by eexists. Qed.
Definition uPred_wand {M} := proj1_sig uPred_wand_aux M.
Definition uPred_wand_eq :
  @uPred_wand = @uPred_wand_def := proj2_sig uPred_wand_aux.

Program Definition uPred_always_def {M} (P : uPred M) : uPred M :=
  {| uPred_holds n x := P n (unit x) |}.
Next Obligation. by intros M P x1 x2 n ? Hx; rewrite /= -Hx. Qed.
Next Obligation.
  intros M P n1 n2 x1 x2 ????; eapply uPred_weaken with n1 (unit x1);
    eauto using cmra_unit_preserving, cmra_unit_validN.
Qed.
Definition uPred_always_aux : { x | x = @uPred_always_def }. by eexists. Qed.
Definition uPred_always {M} := proj1_sig uPred_always_aux M.
Definition uPred_always_eq :
  @uPred_always = @uPred_always_def := proj2_sig uPred_always_aux.

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 [|n] ??; eauto using uPred_ne,(dist_le (A:=M)). Qed.
Next Obligation.
  intros M P [|n1] [|n2] x1 x2; eauto using uPred_weaken,cmra_validN_S; try lia.
Qed.
Definition uPred_later_aux : { x | x = @uPred_later_def }. by eexists. Qed.
Definition uPred_later {M} := proj1_sig uPred_later_aux M.
Definition uPred_later_eq :
  @uPred_later = @uPred_later_def := proj2_sig uPred_later_aux.

Program Definition uPred_ownM_def {M : cmraT} (a : M) : uPred M :=
  {| uPred_holds n x := a ≼{n} x |}.
Next Obligation. by intros M a n x1 x2 [a' ?] Hx; exists a'; rewrite -Hx. Qed.
Next Obligation.
  intros M a n1 n2 x1 x [a' Hx1] [x2 Hx] ??.
  exists (a' ⋅ x2). by rewrite (assoc op) -(dist_le _ _ _ _ Hx1) // Hx.
Qed.
Definition uPred_ownM_aux : { x | x = @uPred_ownM_def }. by eexists. Qed.
Definition uPred_ownM {M} := proj1_sig uPred_ownM_aux M.
Definition uPred_ownM_eq :
  @uPred_ownM = @uPred_ownM_def := proj2_sig uPred_ownM_aux.

Program Definition uPred_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_valid_aux : { x | x = @uPred_valid_def }. by eexists. Qed.
Definition uPred_valid {M A} := proj1_sig uPred_valid_aux M A.
Definition uPred_valid_eq :
  @uPred_valid = @uPred_valid_def := proj2_sig uPred_valid_aux.

Notation "P ⊑ Q" := (uPred_entails P%I Q%I) (at level 70) : C_scope.
Notation "(⊑)" := uPred_entails (only parsing) : C_scope.
Notation "■ φ" := (uPred_const φ%C%type)
  (at level 20, right associativity) : uPred_scope.
Notation "x = y" := (uPred_const (x%C%type = y%C%type)) : uPred_scope.
Notation "'False'" := (uPred_const False) : uPred_scope.
Notation "'True'" := (uPred_const True) : uPred_scope.
Infix "∧" := uPred_and : uPred_scope.
Notation "(∧)" := uPred_and (only parsing) : uPred_scope.
Infix "∨" := uPred_or : uPred_scope.
Notation "(∨)" := uPred_or (only parsing) : uPred_scope.
Infix "→" := uPred_impl : uPred_scope.
Infix "★" := uPred_sep (at level 80, right associativity) : uPred_scope.
Notation "(★)" := uPred_sep (only parsing) : uPred_scope.
Notation "P -★ Q" := (uPred_wand P Q)
  (at level 199, Q at level 200, right associativity) : uPred_scope.
Notation "∀ x .. y , P" :=
  (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I) : uPred_scope.
Notation "∃ x .. y , P" :=
  (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I) : uPred_scope.
Notation "□ P" := (uPred_always P)
  (at level 20, right associativity) : uPred_scope.
Notation "▷ P" := (uPred_later P)
  (at level 20, right associativity) : uPred_scope.
Infix "≡" := uPred_eq : uPred_scope.
Notation "✓ x" := (uPred_valid x) (at level 20) : uPred_scope.

Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P → Q) ∧ (Q → P))%I.
Infix "↔" := uPred_iff : uPred_scope.

Class TimelessP {M} (P : uPred M) := timelessP : ▷ P ⊑ (P ∨ ▷ False).
Arguments timelessP {_} _ {_}.
Class AlwaysStable {M} (P : uPred M) := always_stable : P ⊑ □ P.
Arguments always_stable {_} _ {_}.

Module uPred.
Definition unseal :=
  (uPred_const_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
  uPred_exist_eq, uPred_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq,
  uPred_later_eq, uPred_ownM_eq, uPred_valid_eq).
Ltac unseal := rewrite !unseal.

Section uPred_logic.
Context {M : cmraT}.
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 *)
Arguments uPred_holds {_} !_ _ _ /.
Hint Immediate uPred_in_entails.

Global Instance: PreOrder (@uPred_entails M).
Proof.
  split.
  * by intros P; split=> x i.
  * by intros P Q Q' HP HQ; split=> x i ??; apply HQ, HP.
Qed.
Global Instance: AntiSymm (≡) (@uPred_entails M).
Proof. intros P Q HPQ HQP; split=> x n; by split; [apply HPQ|apply HQP]. Qed.
Lemma equiv_spec P Q : P ≡ Q ↔ P ⊑ Q ∧ Q ⊑ P.
Proof.
  split; [|by intros [??]; apply (anti_symm (⊑))].
  intros HPQ; split; split=> x i; apply HPQ.
Qed.
Lemma equiv_entails P Q : P ≡ Q → P ⊑ Q.
Proof. apply equiv_spec. Qed.
Lemma equiv_entails_sym P Q : Q ≡ P → P ⊑ Q.
Proof. apply equiv_spec. Qed.
Global Instance entails_proper :
  Proper ((≡) ==> (≡) ==> iff) ((⊑) : relation (uPred M)).
Proof.
  move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split; intros.
  - by trans P1; [|trans Q1].
  - by trans P2; [|trans Q2].
Qed.
Lemma entails_equiv_l (P Q R : uPred M) : P ≡ Q → Q ⊑ R → P ⊑ R.
Proof. by intros ->. Qed.
Lemma entails_equiv_r (P Q R : uPred M) : P ⊑ Q → Q ≡ R → P ⊑ R.
Proof. by intros ? <-. Qed.

(** Non-expansiveness and setoid morphisms *)
Global Instance const_proper : Proper (iff ==> (≡)) (@uPred_const M).
Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed.
Global Instance and_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
Proof.
  intros P P' HP Q Q' HQ; unseal; split=> x n' ??.
  split; (intros [??]; split; [by apply HP|by apply HQ]).
Qed.
Global Instance and_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_and M) := ne_proper_2 _.
Global Instance or_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
Proof.
  intros P P' HP Q Q' HQ; split=> x n' ??.
  unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]).
Qed.
Global Instance or_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_or M) := ne_proper_2 _.
Global Instance impl_ne n :
  Proper (dist n ==> dist n ==> dist n) (@uPred_impl M).
Proof.
  intros P P' HP Q Q' HQ; split=> x n' ??.
  unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto.
Qed.
Global Instance impl_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_impl M) := ne_proper_2 _.
Global Instance sep_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
Proof.
  intros P P' HP Q Q' HQ; split=> n' x ??.
  unseal; split; intros (x1&x2&?&?&?); cofe_subst x;
    exists x1, x2; split_and!; try (apply HP || apply HQ);
    eauto using cmra_validN_op_l, cmra_validN_op_r.
Qed.
Global Instance sep_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_sep M) := ne_proper_2 _.
Global Instance wand_ne n :
  Proper (dist n ==> dist n ==> dist n) (@uPred_wand M).
Proof.
  intros 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.
Qed.
Global Instance wand_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_wand M) := ne_proper_2 _.
Global Instance eq_ne (A : cofeT) n :
  Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
Proof.
  intros 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.
Qed.
Global Instance eq_proper (A : cofeT) :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_eq M A) := ne_proper_2 _.
Global Instance forall_ne A n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
Global Instance forall_proper A :
  Proper (pointwise_relation _ (≡) ==> (≡)) (@uPred_forall M A).
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
Global Instance exist_ne A n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
Proof.
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ.
Qed.
Global Instance exist_proper A :
  Proper (pointwise_relation _ (≡) ==> (≡)) (@uPred_exist M A).
Proof.
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ?; split; intros [a ?]; exists a; by apply HΨ.
Qed.
Global Instance later_contractive : Contractive (@uPred_later M).
Proof.
  intros n P Q HPQ; unseal; split=> -[|n'] x ??; simpl; [done|].
  apply (HPQ n'); eauto using cmra_validN_S.
Qed.
Global Instance later_proper :
  Proper ((≡) ==> (≡)) (@uPred_later M) := ne_proper _.
Global Instance always_ne n : Proper (dist n ==> dist n) (@uPred_always M).
Proof.
  intros P1 P2 HP.
  unseal; split=> n' x; split; apply HP; eauto using cmra_unit_validN.
Qed.
Global Instance always_proper :
  Proper ((≡) ==> (≡)) (@uPred_always M) := ne_proper _.
Global Instance ownM_ne n : Proper (dist n ==> dist n) (@uPred_ownM M).
Proof.
  intros 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 valid_ne {A : cmraT} n :
Proper (dist n ==> dist n) (@uPred_valid M A).
Proof.
  intros a b Ha; unseal; split=> n' x ? /=.
  by rewrite (dist_le _ _ _ _ Ha); last lia.
Qed.
Global Instance valid_proper {A : cmraT} :
  Proper ((≡) ==> (≡)) (@uPred_valid M A) := ne_proper _.
Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
Proof. unfold uPred_iff; solve_proper. Qed.
Global Instance iff_proper :
  Proper ((≡) ==> (≡) ==> (≡)) (@uPred_iff M) := ne_proper_2 _.

(** Introduction and elimination rules *)
Lemma const_intro φ P : φ → P ⊑ ■ φ.
Proof. by intros ?; unseal; split. Qed.
Lemma const_elim φ Q R : Q ⊑ ■ φ → (φ → Q ⊑ R) → Q ⊑ R.
Proof.
  unseal; intros HQP HQR; split=> n x ??; apply HQR; first eapply HQP; eauto.
Qed.
Lemma False_elim P : False ⊑ P.
Proof. by unseal; split=> n x ?. Qed.
Lemma and_elim_l P Q : (P ∧ Q) ⊑ P.
Proof. by unseal; split=> n x ? [??]. Qed.
Lemma and_elim_r P Q : (P ∧ Q) ⊑ Q.
Proof. by unseal; split=> n x ? [??]. Qed.
Lemma and_intro P Q R : P ⊑ Q → P ⊑ R → P ⊑ (Q ∧ R).
Proof. intros HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. Qed.
Lemma or_intro_l P Q : P ⊑ (P ∨ Q).
Proof. unseal; split=> n x ??; left; auto. Qed.
Lemma or_intro_r P Q : Q ⊑ (P ∨ Q).
Proof. unseal; split=> n x ??; right; auto. Qed.
Lemma or_elim P Q R : P ⊑ R → Q ⊑ R → (P ∨ Q) ⊑ R.
Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed.
Lemma impl_intro_r P Q R : (P ∧ Q) ⊑ R → P ⊑ (Q → R).
Proof.
  unseal; intros HQ; split=> n x ?? n' x' ????.
  apply HQ; naive_solver eauto using uPred_weaken.
Qed.
Lemma impl_elim P Q R : P ⊑ (Q → R) → P ⊑ Q → P ⊑ R.
Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed.
Lemma forall_intro {A} P (Ψ : A → uPred M): (∀ a, P ⊑ Ψ a) → P ⊑ (∀ a, Ψ a).
Proof. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. Qed.
Lemma forall_elim {A} {Ψ : A → uPred M} a : (∀ a, Ψ a) ⊑ Ψ a.
Proof. unseal; split=> n x ? HP; apply HP. Qed.
Lemma exist_intro {A} {Ψ : A → uPred M} a : Ψ a ⊑ (∃ a, Ψ a).
Proof. unseal; split=> n x ??; by exists a. Qed.
Lemma exist_elim {A} (Φ : A → uPred M) Q : (∀ a, Φ a ⊑ Q) → (∃ a, Φ a) ⊑ Q.
Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed.
Lemma eq_refl {A : cofeT} (a : A) P : P ⊑ (a ≡ a).
Proof. unseal; by split=> n x ??; simpl. Qed.
Lemma eq_rewrite {A : cofeT} a b (Ψ : A → uPred M) P
  `{HΨ : ∀ n, Proper (dist n ==> dist n) Ψ} : P ⊑ (a ≡ b) → P ⊑ Ψ a → P ⊑ Ψ b.
Proof.
  unseal; intros Hab Ha; split=> n x ??.
  apply HΨ with n a; auto. by symmetry; apply Hab with x. by apply Ha.
Qed.
Lemma eq_equiv `{Empty M, !CMRAIdentity M} {A : cofeT} (a b : A) :
  True ⊑ (a ≡ b) → a ≡ b.
Proof.
  unseal=> Hab; apply equiv_dist; intros n; apply Hab with ∅; last done.
  apply cmra_valid_validN, cmra_empty_valid.
Qed.
Lemma iff_equiv P Q : True ⊑ (P ↔ Q) → P ≡ Q.
Proof.
  rewrite /uPred_iff; unseal=> HPQ.
  split=> n x ?; split; intros; by apply HPQ with n x.
Qed.

(* Derived logical stuff *)
Lemma True_intro P : P ⊑ True.
Proof. by apply const_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 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.
Lemma impl_entails P Q : True ⊑ (P → Q) → P ⊑ Q.
Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed.
Lemma entails_impl P Q : (P ⊑ Q) → True ⊑ (P → Q).
Proof. auto using impl_intro_l. Qed.

Lemma const_mono φ1 φ2 : (φ1 → φ2) → ■ φ1 ⊑ ■ φ2.
Proof. intros; apply const_elim with φ1; eauto using const_intro. Qed.
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 const_mono' : Proper (impl ==> (⊑)) (@uPred_const M).
Proof. intros φ1 φ2; apply const_mono. 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.