logic.v 32.8 KB
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Require Export modures.cmra.
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Local Hint Extern 1 (_  _) => etransitivity; [eassumption|].
Local Hint Extern 1 (_  _) => etransitivity; [|eassumption].
Local Hint Extern 10 (_  _) => omega.

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Record uPred (M : cmraT) : Type := IProp {
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  uPred_holds :> nat  M  Prop;
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  uPred_ne x1 x2 n : uPred_holds n x1  x1 ={n}= x2  uPred_holds n x2;
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  uPred_0 x : uPred_holds 0 x;
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  uPred_weaken x1 x2 n1 n2 :
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    uPred_holds n1 x1  x1  x2  n2  n1  {n2} x2  uPred_holds n2 x2
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}.
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Arguments uPred_holds {_} _ _ _.
Hint Resolve uPred_0.
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Add Printing Constructor uPred.
Instance: Params (@uPred_holds) 3.
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Instance uPred_equiv (M : cmraT) : Equiv (uPred M) := λ P Q,  x n,
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  {n} x  P n x  Q n x.
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Instance uPred_dist (M : cmraT) : Dist (uPred M) := λ n P Q,  x n',
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  n'  n  {n'} x  P n' x  Q n' x.
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Program Instance uPred_compl (M : cmraT) : Compl (uPred M) := λ c,
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  {| uPred_holds n x := c n n x |}.
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Next Obligation. by intros M c x y n ??; simpl in *; apply uPred_ne with x. Qed.
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Next Obligation. by intros M c x; simpl. Qed.
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Next Obligation.
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  intros M c x1 x2 n1 n2 ????; simpl in *.
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  apply (chain_cauchy c n2 n1); eauto using uPred_weaken.
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Qed.
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Instance uPred_cofe (M : cmraT) : Cofe (uPred M).
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Proof.
  split.
  * intros P Q; split; [by intros HPQ n x i ??; apply HPQ|].
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    intros HPQ x n ?; apply HPQ with n; auto.
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  * 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.
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  * intros P Q x i; rewrite Nat.le_0_r; intros ->; split; intros; apply uPred_0.
  * by intros c n x i ??; apply (chain_cauchy c i n).
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Qed.
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Instance uPred_holds_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_holds_proper {M} (P : uPred M) n : Proper (() ==> iff) (P n).
Proof. by intros x1 x2 Hx; apply uPred_holds_ne, equiv_dist. Qed.
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Canonical Structure uPredC (M : cmraT) : cofeT := CofeT (uPred M).
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(** functor *)
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Program Definition uPred_map {M1 M2 : cmraT} (f : M2  M1)
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  `{! n, Proper (dist n ==> dist n) f, !CMRAMonotone f}
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  (P : uPred M1) : uPred M2 := {| uPred_holds n x := P n (f x) |}.
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Next Obligation. by intros M1 M2 f ?? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed.
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Next Obligation. intros M1 M2 f _ _ P x; apply uPred_0. Qed.
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Next Obligation.
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  naive_solver eauto using uPred_weaken, included_preserving, validN_preserving.
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Qed.
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Instance uPred_map_ne {M1 M2 : cmraT} (f : M2  M1)
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  `{! n, Proper (dist n ==> dist n) f, !CMRAMonotone f} :
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  Proper (dist n ==> dist n) (uPred_map f).
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Proof.
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  by intros n x1 x2 Hx y n'; split; apply Hx; auto using validN_preserving.
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Qed.
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Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} :
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  uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1  uPredC M2).
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Lemma upredC_map_ne {M1 M2 : cmraT} (f g : M2 -n> M1)
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    `{!CMRAMonotone f, !CMRAMonotone g} n :
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  f ={n}= g  uPredC_map f ={n}= uPredC_map g.
Proof.
  by intros Hfg P y n' ??; simpl; rewrite (dist_le _ _ _ _(Hfg y)) by lia.
Qed.
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(** logical entailement *)
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Instance uPred_entails {M} : SubsetEq (uPred M) := λ P Q,  x n,
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  {n} x  P n x  Q n x.
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(** logical connectives *)
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Program Definition uPred_const {M} (P : Prop) : uPred M :=
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  {| uPred_holds n x := match n return _ with 0 => True | _ => P end |}.
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Solve Obligations with done.
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Next Obligation. intros M P x1 x2 [|n1] [|n2]; auto with lia. Qed.
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Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_const True).
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Program Definition uPred_and {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.
Program Definition uPred_or {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.
Program Definition uPred_impl {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  x' n',
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       x  x'  n'  n  {n'} x'  P n' x'  Q n' x' |}.
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Next Obligation.
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  intros M P Q x1' x1 n1 HPQ Hx1 x2 n2 ????.
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  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).
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  assert ({n2} x2') by (by rewrite Hx2'); rewrite <-Hx2'.
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  eauto using uPred_weaken, uPred_ne.
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Qed.
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Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed.
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Next Obligation. naive_solver eauto 2 with lia. Qed.

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Program Definition uPred_forall {M A} (P : A  uPred M) : uPred M :=
  {| uPred_holds n x :=  a, P a n x |}.
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
Program Definition uPred_exist {M A} (P : A  uPred M) : uPred M :=
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  {| uPred_holds n x :=
       match n return _ with 0 => True | _ =>  a, P a n x end |}.
Next Obligation. intros M A P x y [|n]; naive_solver eauto using uPred_ne. Qed.
Next Obligation. done. Qed.
Next Obligation.
  intros M A P x y [|n] [|n']; naive_solver eauto 2 using uPred_weaken with lia.
Qed.
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Program Definition uPred_eq {M} {A : cofeT} (a1 a2 : A) : uPred M :=
  {| uPred_holds n x := a1 ={n}= a2 |}.
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Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
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Program Definition uPred_sep {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  x1 x2, x ={n}= x1  x2  P n x1  Q n x2 |}.
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Next Obligation.
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  by intros M P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy.
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Qed.
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Next Obligation. by intros M P Q x; exists x, x. Qed.
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Next Obligation.
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  intros M P Q x y n1 n2 (x1&x2&Hx&?&?) Hxy ? Hvalid.
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  assert ( x2', y ={n2}= x1  x2'  x2  x2') as (x2'&Hy&?).
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  { destruct Hxy as [z Hy]; exists (x2  z); split; eauto using ra_included_l.
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    apply dist_le with n1; auto. by rewrite (associative op), <-Hx, Hy. }
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  rewrite Hy in Hvalid; exists x1, x2'; split_ands; auto.
  * apply uPred_weaken with x1 n1; eauto using cmra_valid_op_l.
  * apply uPred_weaken with x2 n1; eauto using cmra_valid_op_r.
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Qed.

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Program Definition uPred_wand {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  x' n',
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       n'  n  {n'} (x  x')  P n' x'  Q n' (x  x') |}.
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Next Obligation.
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  intros M P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *.
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  rewrite <-(dist_le _ _ _ _ Hx) by done; apply HPQ; auto.
  by rewrite (dist_le _ _ _ n2 Hx).
Qed.
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Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed.
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Next Obligation.
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  intros M P Q x1 x2 n1 n2 HPQ ??? x3 n3 ???; simpl in *.
  apply uPred_weaken with (x1  x3) n3;
    eauto using cmra_valid_included, ra_preserving_r.
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Qed.

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Program Definition uPred_later {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.
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Next Obligation. done. Qed.
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Next Obligation.
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  intros M P x1 x2 [|n1] [|n2]; eauto using uPred_weaken, cmra_valid_S.
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Qed.
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Program Definition uPred_always {M} (P : uPred M) : uPred M :=
  {| uPred_holds n x := P n (unit x) |}.
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Next Obligation. by intros M P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed.
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Next Obligation. by intros; simpl. Qed.
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Next Obligation.
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  intros M P x1 x2 n1 n2 ????; eapply uPred_weaken with (unit x1) n1;
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    auto using ra_unit_preserving, cmra_unit_valid.
Qed.

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Program Definition uPred_own {M : cmraT} (a : M) : uPred M :=
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  {| uPred_holds n x := a {n} x |}.
Next Obligation. by intros M a x1 x2 n [a' ?] Hx; exists a'; rewrite <-Hx. Qed.
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Next Obligation. by intros M a x; exists x. Qed.
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Next Obligation.
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  intros M a x1 x n1 n2 [a' Hx1] [x2 Hx] ??.
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  exists (a'  x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx.
Qed.
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Program Definition uPred_valid {M A : cmraT} (a : A) : uPred M :=
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  {| uPred_holds n x := {n} a |}.
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Solve Obligations with naive_solver eauto 2 using cmra_valid_le, cmra_valid_0.
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Delimit Scope uPred_scope with I.
Bind Scope uPred_scope with uPred.
Arguments uPred_holds {_} _%I _ _.
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Notation "'False'" := (uPred_const False) : uPred_scope.
Notation "'True'" := (uPred_const True) : uPred_scope.
Infix "∧" := uPred_and : uPred_scope.
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Notation "(∧)" := uPred_and (only parsing) : uPred_scope.
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Infix "∨" := uPred_or : uPred_scope.
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Notation "(∨)" := uPred_or (only parsing) : uPred_scope.
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Infix "→" := uPred_impl : uPred_scope.
Infix "★" := uPred_sep (at level 80, right associativity) : uPred_scope.
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Notation "(★)" := uPred_sep (only parsing) : uPred_scope.
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Notation "P -★ Q" := (uPred_wand P Q)
  (at level 90, Q at level 200, right associativity) : uPred_scope.
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Notation "∀ x .. y , P" :=
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  (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I) : uPred_scope.
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Notation "∃ x .. y , P" :=
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  (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I) : uPred_scope.
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Notation "▷ P" := (uPred_later P) (at level 20) : uPred_scope.
Notation "□ P" := (uPred_always P) (at level 20) : uPred_scope.
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Infix "≡" := uPred_eq : uPred_scope.
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Notation "✓" := uPred_valid (at level 1) : uPred_scope.
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Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P  Q)  (Q  P))%I.
Infix "↔" := uPred_iff : uPred_scope.

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Fixpoint uPred_big_and {M} (Ps : list (uPred M)) :=
  match Ps with [] => True | P :: Ps => P  uPred_big_and Ps end%I.
Fixpoint uPred_big_sep {M} (Ps : list (uPred M)) :=
  match Ps with [] => True | P :: Ps => P  uPred_big_sep Ps end%I.

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Class TimelessP {M} (P : uPred M) := timelessP x n : {1} x  P 1 x  P n x.
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Module uPred. Section uPred_logic.
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Context {M : cmraT}.
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Implicit Types P Q : uPred M.
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Implicit Types A : Type.
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Global Instance: PreOrder (() : relation (uPred M)).
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Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
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Global Instance: AntiSymmetric () (() : relation (uPred M)).
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Proof. intros P Q HPQ HQP; split; auto. Qed.
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Lemma equiv_spec P Q : P  Q  P  Q  Q  P.
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Proof.
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  split; [|by intros [??]; apply (anti_symmetric ())].
  intros HPQ; split; intros x i; apply HPQ.
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Qed.
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Global Instance entails_proper :
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  Proper (() ==> () ==> iff) (() : relation (uPred M)).
Proof.
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  intros P1 P2 HP Q1 Q2 HQ; rewrite equiv_spec in HP, HQ; split; intros.
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  * by rewrite (proj2 HP), <-(proj1 HQ).
  * by rewrite (proj1 HP), <-(proj2 HQ).
Qed.
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(** Non-expansiveness and setoid morphisms *)
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Global Instance const_proper : Proper (iff ==> ()) (@uPred_const M).
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Proof. by intros P Q HPQ ? [|n] ?; try apply HPQ. Qed.
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Global Instance and_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
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Proof.
  intros P P' HP Q Q' HQ; split; intros [??]; split; by apply HP || by apply HQ.
Qed.
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Global Instance and_proper :
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  Proper (() ==> () ==> ()) (@uPred_and M) := ne_proper_2 _.
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Global Instance or_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
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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 or_proper :
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  Proper (() ==> () ==> ()) (@uPred_or M) := ne_proper_2 _.
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Global Instance impl_ne n :
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  Proper (dist n ==> dist n ==> dist n) (@uPred_impl M).
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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 impl_proper :
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  Proper (() ==> () ==> ()) (@uPred_impl M) := ne_proper_2 _.
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Global Instance sep_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
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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 sep_proper :
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  Proper (() ==> () ==> ()) (@uPred_sep M) := ne_proper_2 _.
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Global Instance wand_ne n :
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  Proper (dist n ==> dist n ==> dist n) (@uPred_wand M).
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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 wand_proper :
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  Proper (() ==> () ==> ()) (@uPred_wand M) := ne_proper_2 _.
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Global Instance eq_ne (A : cofeT) n :
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  Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
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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 eq_proper (A : cofeT) :
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  Proper (() ==> () ==> ()) (@uPred_eq M A) := ne_proper_2 _.
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Global Instance forall_ne A :
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  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
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Proof. by intros n P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
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Global Instance forall_proper A :
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  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
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Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
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Global Instance exists_ne A :
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  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
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Proof.
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  by intros n P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
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Qed.
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Global Instance exist_proper A :
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  Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
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Proof.
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  by intros P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
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Qed.
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Global Instance later_contractive : Contractive (@uPred_later M).
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Proof.
  intros n P Q HPQ x [|n'] ??; simpl; [done|].
  apply HPQ; eauto using cmra_valid_S.
Qed.
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Global Instance later_proper :
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  Proper (() ==> ()) (@uPred_later M) := ne_proper _.
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Global Instance always_ne n: Proper (dist n ==> dist n) (@uPred_always M).
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Proof. intros P1 P2 HP x n'; split; apply HP; eauto using cmra_unit_valid. Qed.
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Global Instance always_proper :
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  Proper (() ==> ()) (@uPred_always M) := ne_proper _.
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Global Instance own_ne n : Proper (dist n ==> dist n) (@uPred_own M).
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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.
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Global Instance own_proper : Proper (() ==> ()) (@uPred_own M) := ne_proper _.
Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
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Proof. unfold uPred_iff; solve_proper. Qed.
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Global Instance iff_proper :
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  Proper (() ==> () ==> ()) (@uPred_iff M) := ne_proper_2 _.
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(** Introduction and elimination rules *)
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Lemma const_intro P (Q : Prop) : Q  P  uPred_const Q.
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Proof. by intros ?? [|?]. Qed.
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Lemma const_elim (P : Prop) Q R : (P  Q  R)  (Q  uPred_const P)%I  R.
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Proof. intros HR x [|n] ? [??]; [|apply HR]; auto. Qed.
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Lemma True_intro P : P  True%I.
Proof. by apply const_intro. Qed.
Lemma False_elim P : False%I  P.
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Proof. by intros x [|n] ?. Qed.
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Lemma and_elim_l P Q : (P  Q)%I  P.
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Proof. by intros x n ? [??]. Qed.
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Lemma and_elim_r P Q : (P  Q)%I  Q.
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Proof. by intros x n ? [??]. Qed.
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Lemma and_intro P Q R : P  Q  P  R  P  (Q  R)%I.
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Proof. intros HQ HR x n ??; split; auto. Qed.
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Lemma or_intro_l P Q : P  (P  Q)%I.
Proof. intros x n ??; left; auto. Qed.
Lemma or_intro_r P Q : Q  (P  Q)%I.
Proof. intros x n ??; right; auto. Qed.
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Lemma or_elim R P Q : P  R  Q  R  (P  Q)%I  R.
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Proof. intros HP HQ x n ? [?|?]. by apply HP. by apply HQ. Qed.
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Lemma impl_intro P Q R : (R  P)%I  Q  R  (P  Q)%I.
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Proof.
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  intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using uPred_weaken.
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Qed.
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Lemma impl_elim P Q R : P  (Q  R)%I  P  Q  P  R.
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Proof. by intros HP HP' x n ??; apply HP with x n, HP'. Qed.
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Lemma forall_intro P `(Q: A  uPred M): ( a, P  Q a)  P  ( a, Q a)%I.
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Proof. by intros HPQ x n ?? a; apply HPQ. Qed.
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Lemma forall_elim `(P : A  uPred M) a : ( a, P a)%I  P a.
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Proof. intros x n ? HP; apply HP. Qed.
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Lemma exist_intro `(P : A  uPred M) a : P a  ( a, P a)%I.
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Proof. by intros x [|n] ??; [done|exists a]. Qed.
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Lemma exist_elim `(P : A  uPred M) Q : ( a, P a  Q)  ( a, P a)%I  Q.
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Proof. by intros HPQ x [|n] ?; [|intros [a ?]; apply HPQ with a]. Qed.
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Lemma eq_refl {A : cofeT} (a : A) P : P  (a  a)%I.
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Proof. by intros x n ??; simpl. Qed.
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Lemma eq_rewrite {A : cofeT} P (Q : A  uPred M)
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    `{HQ : ! n, Proper (dist n ==> dist n) Q} a b :
  P  (a  b)%I  P  Q a  P  Q b.
Proof.
  intros Hab Ha x n ??; apply HQ with n a; auto. by symmetry; apply Hab with x.
Qed.
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Lemma eq_equiv `{Empty M, !RAIdentity M} {A : cofeT} (a b : A) :
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  True%I  (a  b : uPred M)%I  a  b.
Proof.
  intros Hab; apply equiv_dist; intros n; apply Hab with .
  * apply cmra_valid_validN, ra_empty_valid.
  * by destruct n.
Qed.
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Lemma iff_equiv P Q : True%I  (P  Q)%I  P  Q.
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Proof. by intros HPQ x [|n] ?; [|split; intros; apply HPQ with x (S n)]. Qed.

(* Derived logical stuff *)
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Lemma and_elim_l' P Q R : P  R  (P  Q)%I  R.
Proof. by rewrite and_elim_l. Qed.
Lemma and_elim_r' P Q R : Q  R  (P  Q)%I  R.
Proof. by rewrite and_elim_r. Qed.
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Lemma or_intro_l' P Q R : P  Q  P  (Q  R)%I.
Proof. intros ->; apply or_intro_l. Qed.
Lemma or_intro_r' P Q R : P  R  P  (Q  R)%I.
Proof. intros ->; apply or_intro_r. Qed.
Lemma exist_intro' P `(Q : A  uPred M) a : P  Q a  P  ( a, Q a)%I.
Proof. intros ->; apply exist_intro. Qed.
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Hint Resolve or_elim or_intro_l' or_intro_r'.
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Hint Resolve and_intro and_elim_l' and_elim_r'.
Hint Immediate True_intro False_elim.
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Global Instance and_idem : Idempotent () (@uPred_and M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance or_idem : Idempotent () (@uPred_or M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance and_comm : Commutative () (@uPred_and M).
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Proof. intros P Q; apply (anti_symmetric ()); auto. Qed.
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Global Instance True_and : LeftId () True%I (@uPred_and M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance and_True : RightId () True%I (@uPred_and M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance False_and : LeftAbsorb () False%I (@uPred_and M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance and_False : RightAbsorb () False%I (@uPred_and M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance True_or : LeftAbsorb () True%I (@uPred_or M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance or_True : RightAbsorb () True%I (@uPred_or M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance False_or : LeftId () False%I (@uPred_or M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance or_False : RightId () False%I (@uPred_or M).
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Proof. intros P; apply (anti_symmetric ()); auto. Qed.
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Global Instance and_assoc : Associative () (@uPred_and M).
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Proof. intros P Q R; apply (anti_symmetric ()); auto. Qed.
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Global Instance or_comm : Commutative () (@uPred_or M).
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Proof. intros P Q; apply (anti_symmetric ()); auto. Qed.
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Global Instance or_assoc : Associative () (@uPred_or M).
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Proof. intros P Q R; apply (anti_symmetric ()); auto. Qed.

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Lemma const_mono (P Q: Prop) : (P  Q)  uPred_const P  @uPred_const M Q.
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Proof.
  intros; rewrite <-(left_id True%I _ (uPred_const P)).
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  eauto using const_elim, const_intro.
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Qed.
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Lemma and_mono P P' Q Q' : P  Q  P'  Q'  (P  P')%I  (Q  Q')%I.
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Proof. auto. Qed.
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Lemma or_mono P P' Q Q' : P  Q  P'  Q'  (P  P')%I  (Q  Q')%I.
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Proof. auto. Qed.
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Lemma impl_mono P P' Q Q' : Q  P  P'  Q'  (P  P')%I  (Q  Q')%I.
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Proof.
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  intros HP HQ'; apply impl_intro; rewrite <-HQ'.
  transitivity ((P  P')  P)%I; eauto using impl_elim.
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Qed.
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Lemma forall_mono {A} (P Q : A  uPred M) :
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  ( a, P a  Q a)  ( a, P a)%I  ( a, Q a)%I.
Proof.
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  intros HPQ. apply forall_intro; intros a; rewrite <-(HPQ a).
  apply forall_elim.
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Qed.
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Lemma exist_mono {A} (P Q : A  uPred M) :
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  ( a, P a  Q a)  ( a, P a)%I  ( a, Q a)%I.
Proof.
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  intros HPQ. apply exist_elim; intros a; rewrite (HPQ a); apply exist_intro.
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Qed.

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Global Instance const_mono' : Proper (impl ==> ()) (@uPred_const M).
Proof. intros P Q; 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 or_mono' : Proper (() ==> () ==> ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance impl_mono' :
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  Proper (flip () ==> () ==> ()) (@uPred_impl M).
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Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
Global Instance forall_mono' A :
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  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
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Proof. intros P1 P2; apply forall_mono. Qed.
Global Instance exist_mono' A :
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  Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
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Proof. intros P1 P2; apply exist_mono. Qed.
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Lemma equiv_eq {A : cofeT} P (a b : A) : a  b  P  (a  b)%I.
Proof. intros ->; apply eq_refl. Qed.
Lemma eq_sym {A : cofeT} (a b : A) : (a  b)%I  (b  a : uPred M)%I.
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Proof.
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  refine (eq_rewrite _ (λ b, b  a)%I a b _ _); auto using eq_refl.
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  intros n; solve_proper.
Qed.
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(* BI connectives *)
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Lemma sep_mono P P' Q Q' : P  Q  P'  Q'  (P  P')%I  (Q  Q')%I.
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Proof.
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  intros HQ HQ' x n' Hx' (x1&x2&Hx&?&?); exists x1, x2;
    rewrite Hx in Hx'; eauto 7 using cmra_valid_op_l, cmra_valid_op_r.
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Qed.
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Global Instance True_sep : LeftId () True%I (@uPred_sep M).
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Proof.
  intros P x n Hvalid; split.
  * intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
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    eauto using uPred_weaken, ra_included_r.
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  * by destruct n as [|n]; [|intros ?; exists (unit x), x; rewrite ra_unit_l].
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Qed. 
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Global Instance sep_commutative : Commutative () (@uPred_sep M).
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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.
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Global Instance sep_associative : Associative () (@uPred_sep M).
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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.
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Lemma wand_intro P Q R : (R  P)%I  Q  R  (P - Q)%I.
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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 uPred_weaken with x n; eauto using cmra_valid_op_l.
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Qed.
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Lemma wand_elim P Q : ((P - Q)  P)%I  Q.
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Proof.
  by intros x n Hvalid (x1&x2&Hx&HPQ&?); rewrite Hx in Hvalid |- *; apply HPQ.
Qed.
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Lemma or_sep_distr P Q R : ((P  Q)  R)%I  ((P  R)  (Q  R))%I.
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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.
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Lemma exist_sep_distr `(P : A  uPred M) Q :
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  (( b, P b)  Q)%I  ( b, P b  Q)%I.
Proof.
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  intros x [|n]; [done|split; [by intros (x1&x2&Hx&[a ?]&?); exists a,x1,x2|]].
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  intros [a (x1&x2&Hx&?&?)]; exists x1, x2; split_ands; by try exists a.
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Qed.
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(* Derived BI Stuff *)
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Hint Resolve sep_mono.
Global Instance sep_mono' : Proper (() ==> () ==> ()) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Lemma wand_mono P P' Q Q' : Q  P  P'  Q'  (P - P')%I  (Q - Q')%I.
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Proof.
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  intros HP HQ; apply wand_intro; rewrite HP, <-HQ; apply wand_elim.
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Qed.
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Global Instance wand_mono' : Proper (flip () ==> () ==> ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
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Global Instance sep_True : RightId () True%I (@uPred_sep M).
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Proof. by intros P; rewrite (commutative _), (left_id _ _). Qed.
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Lemma sep_elim_l P Q : (P  Q)%I  P.
Proof. by rewrite (True_intro Q), (right_id _ _). Qed.
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Lemma sep_elim_r P Q : (P  Q)%I  Q.
Proof. by rewrite (commutative ())%I; apply sep_elim_l. Qed.
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Lemma sep_elim_l' P Q R : P  R  (P  Q)%I  R.
Proof. intros ->; apply sep_elim_l. Qed.
Lemma sep_elim_r' P Q R : Q  R  (P  Q)%I  R.
Proof. intros ->; apply sep_elim_r. Qed.
Hint Resolve sep_elim_l' sep_elim_r'.
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Lemma sep_and P Q : (P  Q)%I  (P  Q)%I.
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Proof. auto. Qed.
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Global Instance sep_False : LeftAbsorb () False%I (@uPred_sep M).
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Proof. intros P; apply (anti_symmetric _); auto. Qed.
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Global Instance False_sep : RightAbsorb () False%I (@uPred_sep M).
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Proof. intros P; apply (anti_symmetric _); auto. Qed.
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Lemma impl_wand P Q : (P  Q)%I  (P - Q)%I.
Proof. apply wand_intro, impl_elim with P; auto. Qed.
Lemma and_sep_distr P Q R : ((P  Q)  R)%I  ((P  R)  (Q  R))%I.
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Proof. auto. Qed.
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Lemma forall_sep_distr `(P : A  uPred M) Q :
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  (( a, P a)  Q)%I  ( a, P a  Q)%I.
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Proof. by apply forall_intro; intros a; rewrite forall_elim. Qed.
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(* Later *)
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Lemma later_mono P Q : P  Q  ( P)%I  ( Q)%I.
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Proof. intros HP x [|n] ??; [done|apply HP; auto using cmra_valid_S]. Qed.
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Lemma later_intro P : P  ( P)%I.
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Proof.
  intros x [|n] ??; simpl in *; [done|].
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  apply uPred_weaken with x (S n); auto using cmra_valid_S.
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Qed.
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Lemma lub P : ( P  P)%I  P.
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Proof.
  intros x n ? HP; induction n as [|n IH]; [by apply HP|].
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  apply HP, IH, uPred_weaken with x (S n); eauto using cmra_valid_S.
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Qed.
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Lemma later_and P Q : ( (P  Q))%I  ( P   Q)%I.
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Proof. by intros x [|n]; split. Qed.
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Lemma later_or P Q : ( (P  Q))%I  ( P   Q)%I.
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Proof. intros x [|n]; simpl; tauto. Qed.
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Lemma later_forall `(P : A  uPred M) : (  a, P a)%I  ( a,  P a)%I.
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Proof. by intros x [|n]. Qed.