Commit 0f025d83 authored by Robbert Krebbers's avatar Robbert Krebbers
Browse files

More consistent names in logic.v.

parent 8f1dfe41
......@@ -3,8 +3,8 @@ Local Hint Extern 1 (_ ≼ _) => etransitivity; [eassumption|].
Local Hint Extern 1 (_ _) => etransitivity; [|eassumption].
Local Hint Extern 10 (_ _) => omega.
Structure iProp (A : cmraT) : Type := IProp {
iprop_holds :> nat A -> Prop;
Structure iProp (M : cmraT) : Type := IProp {
iprop_holds :> nat M -> Prop;
iprop_ne x1 x2 n : iprop_holds n x1 x1 ={n}= x2 iprop_holds n x2;
iprop_weaken x1 x2 n1 n2 :
x1 x2 n2 n1 validN n2 x2 iprop_holds n1 x1 iprop_holds n2 x2
......@@ -12,18 +12,18 @@ Structure iProp (A : cmraT) : Type := IProp {
Add Printing Constructor iProp.
Instance: Params (@iprop_holds) 3.
Instance iprop_equiv (A : cmraT) : Equiv (iProp A) := λ P Q, x n,
Instance iprop_equiv (M : cmraT) : Equiv (iProp M) := λ P Q, x n,
validN n x P n x Q n x.
Instance iprop_dist (A : cmraT) : Dist (iProp A) := λ n P Q, x n',
Instance iprop_dist (M : cmraT) : Dist (iProp M) := λ n P Q, x n',
n' < n validN n' x P n' x Q n' x.
Program Instance iprop_compl (A : cmraT) : Compl (iProp A) := λ c,
Program Instance iprop_compl (M : cmraT) : Compl (iProp M) := λ c,
{| iprop_holds n x := c (S n) n x |}.
Next Obligation. by intros A c x y n ??; simpl in *; apply iprop_ne with x. Qed.
Next Obligation. by intros M c x y n ??; simpl in *; apply iprop_ne with x. Qed.
Next Obligation.
intros A c x1 x2 n1 n2 ????; simpl in *.
intros M c x1 x2 n1 n2 ????; simpl in *.
apply (chain_cauchy c (S n2) (S n1)); eauto using iprop_weaken, cmra_valid_le.
Qed.
Instance iprop_cofe (A : cmraT) : Cofe (iProp A).
Instance iprop_cofe (M : cmraT) : Cofe (iProp M).
Proof.
split.
* intros P Q; split; [by intros HPQ n x i ??; apply HPQ|].
......@@ -36,50 +36,50 @@ Proof.
* intros P Q x i ??; lia.
* intros c n x i ??; apply (chain_cauchy c (S i) n); auto.
Qed.
Instance iprop_holds_ne {A} (P : iProp A) n : Proper (dist n ==> iff) (P n).
Instance iprop_holds_ne {M} (P : iProp M) n : Proper (dist n ==> iff) (P n).
Proof. intros x1 x2 Hx; split; eauto using iprop_ne. Qed.
Instance iprop_holds_proper {A} (P : iProp A) n : Proper (() ==> iff) (P n).
Instance iprop_holds_proper {M} (P : iProp M) n : Proper (() ==> iff) (P n).
Proof. by intros x1 x2 Hx; apply iprop_holds_ne, equiv_dist. Qed.
Definition iPropC (A : cmraT) : cofeT := CofeT (iProp A).
Definition iPropC (M : cmraT) : cofeT := CofeT (iProp M).
(** functor *)
Program Definition iprop_map {A B : cmraT} (f : B A)
Program Definition iprop_map {M1 M2 : cmraT} (f : M2 M1)
`{! n, Proper (dist n ==> dist n) f, !CMRAPreserving f}
(P : iProp A) : iProp B := {| iprop_holds n x := P n (f x) |}.
Next Obligation. by intros A B f ?? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed.
(P : iProp M1) : iProp M2 := {| iprop_holds n x := P n (f x) |}.
Next Obligation. by intros M1 M2 f ?? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed.
Next Obligation.
by intros A B f ?? P y1 y2 n i ???; simpl; apply iprop_weaken; auto;
by intros M1 M2 f ?? P y1 y2 n i ???; simpl; apply iprop_weaken; auto;
apply validN_preserving || apply included_preserving.
Qed.
Instance iprop_map_ne {A B : cmraT} (f : B A)
Instance iprop_map_ne {M1 M2 : cmraT} (f : M2 M1)
`{! n, Proper (dist n ==> dist n) f, !CMRAPreserving f} :
Proper (dist n ==> dist n) (iprop_map f).
Proof.
by intros n x1 x2 Hx y n'; split; apply Hx; try apply validN_preserving.
Qed.
Definition ipropC_map {A B : cmraT} (f : B -n> A) `{!CMRAPreserving f} :
iPropC A -n> iPropC B := CofeMor (iprop_map f : iPropC A iPropC B).
Definition ipropC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAPreserving f} :
iPropC M1 -n> iPropC M2 := CofeMor (iprop_map f : iPropC M1 iPropC M2).
(** logical entailement *)
Instance iprop_entails {A} : SubsetEq (iProp A) := λ P Q, x n,
Instance iprop_entails {M} : SubsetEq (iProp M) := λ P Q, x n,
validN n x P n x Q n x.
(** logical connectives *)
Program Definition iprop_const {A} (P : Prop) : iProp A :=
Program Definition iprop_const {M} (P : Prop) : iProp M :=
{| iprop_holds n x := P |}.
Solve Obligations with done.
Program Definition iprop_and {A} (P Q : iProp A) : iProp A :=
Program Definition iprop_and {M} (P Q : iProp M) : iProp M :=
{| iprop_holds n x := P n x Q n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_or {A} (P Q : iProp A) : iProp A :=
Program Definition iprop_or {M} (P Q : iProp M) : iProp M :=
{| iprop_holds n x := P n x Q n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_impl {A} (P Q : iProp A) : iProp A :=
Program Definition iprop_impl {M} (P Q : iProp M) : iProp M :=
{| iprop_holds n x := x' n',
x x' n' n validN n' x' P n' x' Q n' x' |}.
Next Obligation.
intros A P Q x1' x1 n1 HPQ Hx1 x2 n2 ????.
intros M P Q x1' x1 n1 HPQ Hx1 x2 n2 ????.
destruct (cmra_included_dist_l x1 x2 x1' n1) as (x2'&?&Hx2); auto.
assert (x2' ={n2}= x2) as Hx2' by (by apply dist_le with n1).
assert (validN n2 x2') by (by rewrite Hx2'); rewrite <-Hx2'.
......@@ -87,24 +87,24 @@ Next Obligation.
Qed.
Next Obligation. naive_solver eauto 2 with lia. Qed.
Program Definition iprop_forall {A B} (P : A iProp B) : iProp B :=
Program Definition iprop_forall {M A} (P : A iProp M) : iProp M :=
{| iprop_holds n x := a, P a n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_exist {A B} (P : A iProp B) : iProp B :=
Program Definition iprop_exist {M A} (P : A iProp M) : iProp M :=
{| iprop_holds n x := a, P a n x |}.
Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken.
Program Definition iprop_eq {A} {B : cofeT} (b1 b2 : B) : iProp A :=
{| iprop_holds n x := b1 ={n}= b2 |}.
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=B)).
Program Definition iprop_eq {M} {A : cofeT} (a1 a2 : A) : iProp M :=
{| iprop_holds n x := a1 ={n}= a2 |}.
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
Program Definition iprop_sep {A} (P Q : iProp A) : iProp A :=
Program Definition iprop_sep {M} (P Q : iProp M) : iProp M :=
{| iprop_holds n x := x1 x2, x ={n}= x1 x2 P n x1 Q n x2 |}.
Next Obligation.
by intros A P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy.
by intros M P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy.
Qed.
Next Obligation.
intros A P Q x y n1 n2 Hxy ?? (x1&x2&Hx&?&?).
intros M P Q x y n1 n2 Hxy ?? (x1&x2&Hx&?&?).
assert ( x2', y ={n2}= x1 x2' x2 x2') as (x2'&Hy&?).
{ rewrite ra_included_spec in Hxy; destruct Hxy as [z Hy].
exists (x2 z); split; eauto using ra_included_l.
......@@ -116,49 +116,49 @@ Next Obligation.
by apply cmra_valid_op_r with x1; rewrite <-Hy.
Qed.
Program Definition iprop_wand {A} (P Q : iProp A) : iProp A :=
Program Definition iprop_wand {M} (P Q : iProp M) : iProp M :=
{| iprop_holds n x := x' n',
n' n validN n' (x x') P n' x' Q n' (x x') |}.
Next Obligation.
intros A P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *.
intros M P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *.
rewrite <-(dist_le _ _ _ _ Hx) by done; apply HPQ; auto.
by rewrite (dist_le _ _ _ n2 Hx).
Qed.
Next Obligation.
intros A P Q x1 x2 n1 n2 ??? HPQ x3 n3 ???; simpl in *.
intros M P Q x1 x2 n1 n2 ??? HPQ x3 n3 ???; simpl in *.
apply iprop_weaken with (x1 x3) n3; auto using ra_preserving_r.
apply HPQ; auto.
apply cmra_valid_included with (x2 x3); auto using ra_preserving_r.
Qed.
Program Definition iprop_later {A} (P : iProp A) : iProp A :=
Program Definition iprop_later {M} (P : iProp M) : iProp M :=
{| iprop_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
Next Obligation. intros A P ?? [|n]; eauto using iprop_ne,(dist_le (A:=A)). Qed.
Next Obligation. intros M P ?? [|n]; eauto using iprop_ne,(dist_le (A:=M)). Qed.
Next Obligation.
intros A P x1 x2 [|n1] [|n2] ????; auto with lia.
intros M P x1 x2 [|n1] [|n2] ????; auto with lia.
apply iprop_weaken with x1 n1; eauto using cmra_valid_S.
Qed.
Program Definition iprop_always {A} (P : iProp A) : iProp A :=
Program Definition iprop_always {M} (P : iProp M) : iProp M :=
{| iprop_holds n x := P n (unit x) |}.
Next Obligation. by intros A P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed.
Next Obligation. by intros M P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed.
Next Obligation.
intros A P x1 x2 n1 n2 ????; eapply iprop_weaken with (unit x1) n1;
intros M P x1 x2 n1 n2 ????; eapply iprop_weaken with (unit x1) n1;
auto using ra_unit_preserving, cmra_unit_valid.
Qed.
Program Definition iprop_own {A : cmraT} (a : A) : iProp A :=
Program Definition iprop_own {M : cmraT} (a : M) : iProp M :=
{| iprop_holds n x := a', x ={n}= a a' |}.
Next Obligation. by intros A a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed.
Next Obligation. by intros M a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed.
Next Obligation.
intros A a x1 x n1 n2; rewrite ra_included_spec; intros [x2 Hx] ?? [a' Hx1].
intros M a x1 x n1 n2; rewrite ra_included_spec; intros [x2 Hx] ?? [a' Hx1].
exists (a' x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx.
Qed.
Program Definition iprop_valid {A : cmraT} (a : A) : iProp A :=
Program Definition iprop_valid {M : cmraT} (a : M) : iProp M :=
{| iprop_holds n x := validN n a |}.
Solve Obligations with naive_solver eauto 2 using cmra_valid_le.
Definition iprop_fixpoint {A} (P : iProp A iProp A)
`{!Contractive P} : iProp A := fixpoint P (iprop_const True).
Definition iprop_fixpoint {M} (P : iProp M iProp M)
`{!Contractive P} : iProp M := fixpoint P (iprop_const True).
Delimit Scope iprop_scope with I.
Bind Scope iprop_scope with iProp.
......@@ -179,10 +179,10 @@ Notation "▷ P" := (iprop_later P) (at level 20) : iprop_scope.
Notation "□ P" := (iprop_always P) (at level 20) : iprop_scope.
Section logic.
Context {A : cmraT}.
Implicit Types P Q : iProp A.
Context {M : cmraT}.
Implicit Types P Q : iProp M.
Global Instance iprop_preorder : PreOrder (() : relation (iProp A)).
Global Instance iprop_preorder : PreOrder (() : relation (iProp M)).
Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
Lemma iprop_equiv_spec P Q : P Q P Q Q P.
Proof.
......@@ -192,90 +192,90 @@ Proof.
Qed.
(** Non-expansiveness *)
Global Instance iprop_const_proper : Proper (iff ==> ()) (@iprop_const A).
Global Instance iprop_const_proper : Proper (iff ==> ()) (@iprop_const M).
Proof. intros P Q HPQ ???; apply HPQ. Qed.
Global Instance iprop_and_ne n :
Proper (dist n ==> dist n ==> dist n) (@iprop_and A).
Proper (dist n ==> dist n ==> dist n) (@iprop_and M).
Proof.
intros P P' HP Q Q' HQ; split; intros [??]; split; by apply HP || by apply HQ.
Qed.
Global Instance iprop_and_proper :
Proper (() ==> () ==> ()) (@iprop_and A) := ne_proper_2 _.
Proper (() ==> () ==> ()) (@iprop_and M) := ne_proper_2 _.
Global Instance iprop_or_ne n :
Proper (dist n ==> dist n ==> dist n) (@iprop_or A).
Proper (dist n ==> dist n ==> dist n) (@iprop_or M).
Proof.
intros P P' HP Q Q' HQ; split; intros [?|?];
first [by left; apply HP | by right; apply HQ].
Qed.
Global Instance iprop_or_proper :
Proper (() ==> () ==> ()) (@iprop_or A) := ne_proper_2 _.
Proper (() ==> () ==> ()) (@iprop_or M) := ne_proper_2 _.
Global Instance iprop_impl_ne n :
Proper (dist n ==> dist n ==> dist n) (@iprop_impl A).
Proper (dist n ==> dist n ==> dist n) (@iprop_impl M).
Proof.
intros P P' HP Q Q' HQ; split; intros HPQ x' n'' ????; apply HQ,HPQ,HP; auto.
Qed.
Global Instance iprop_impl_proper :
Proper (() ==> () ==> ()) (@iprop_impl A) := ne_proper_2 _.
Proper (() ==> () ==> ()) (@iprop_impl M) := ne_proper_2 _.
Global Instance iprop_sep_ne n :
Proper (dist n ==> dist n ==> dist n) (@iprop_sep A).
Proper (dist n ==> dist n ==> dist n) (@iprop_sep M).
Proof.
intros P P' HP Q Q' HQ x n' ? Hx'; split; intros (x1&x2&Hx&?&?);
exists x1, x2; rewrite Hx in Hx'; split_ands; try apply HP; try apply HQ;
eauto using cmra_valid_op_l, cmra_valid_op_r.
Qed.
Global Instance iprop_sep_proper :
Proper (() ==> () ==> ()) (@iprop_sep A) := ne_proper_2 _.
Proper (() ==> () ==> ()) (@iprop_sep M) := ne_proper_2 _.
Global Instance iprop_wand_ne n :
Proper (dist n ==> dist n ==> dist n) (@iprop_wand A).
Proper (dist n ==> dist n ==> dist n) (@iprop_wand M).
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.
Global Instance iprop_wand_proper :
Proper (() ==> () ==> ()) (@iprop_wand A) := ne_proper_2 _.
Global Instance iprop_eq_ne {B : cofeT} n :
Proper (dist n ==> dist n ==> dist n) (@iprop_eq A B).
Proper (() ==> () ==> ()) (@iprop_wand M) := ne_proper_2 _.
Global Instance iprop_eq_ne {A : cofeT} n :
Proper (dist n ==> dist n ==> dist n) (@iprop_eq M A).
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.
Global Instance iprop_eq_proper {B : cofeT} :
Proper (() ==> () ==> ()) (@iprop_eq A B) := ne_proper_2 _.
Global Instance iprop_forall_ne {B : cofeT} :
Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_forall B A).
Global Instance iprop_eq_proper {A : cofeT} :
Proper (() ==> () ==> ()) (@iprop_eq M A) := ne_proper_2 _.
Global Instance iprop_forall_ne {A : cofeT} :
Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_forall M A).
Proof. by intros n P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
Global Instance iprop_forall_proper {B : cofeT} :
Proper (pointwise_relation _ () ==> ()) (@iprop_forall B A).
Global Instance iprop_forall_proper {A : cofeT} :
Proper (pointwise_relation _ () ==> ()) (@iprop_forall M A).
Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
Global Instance iprop_exists_ne {B : cofeT} :
Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_exist B A).
Global Instance iprop_exists_ne {A : cofeT} :
Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_exist M A).
Proof.
by intros n P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
Qed.
Global Instance iprop_exist_proper {B : cofeT} :
Proper (pointwise_relation _ () ==> ()) (@iprop_exist B A).
Global Instance iprop_exist_proper {A : cofeT} :
Proper (pointwise_relation _ () ==> ()) (@iprop_exist M A).
Proof.
by intros P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
Qed.
Global Instance iprop_later_contractive : Contractive (@iprop_later A).
Global Instance iprop_later_contractive : Contractive (@iprop_later M).
Proof.
intros n P Q HPQ x [|n'] ??; simpl; [done|].
apply HPQ; eauto using cmra_valid_S.
Qed.
Global Instance iprop_later_proper :
Proper (() ==> ()) (@iprop_later A) := ne_proper _.
Global Instance iprop_always_ne n: Proper (dist n ==> dist n) (@iprop_always A).
Proper (() ==> ()) (@iprop_later M) := ne_proper _.
Global Instance iprop_always_ne n: Proper (dist n ==> dist n) (@iprop_always M).
Proof. intros P1 P2 HP x n'; split; apply HP; eauto using cmra_unit_valid. Qed.
Global Instance iprop_always_proper :
Proper (() ==> ()) (@iprop_always A) := ne_proper _.
Global Instance iprop_own_ne n : Proper (dist n ==> dist n) (@iprop_own A).
Proper (() ==> ()) (@iprop_always M) := ne_proper _.
Global Instance iprop_own_ne n : Proper (dist n ==> dist n) (@iprop_own M).
Proof.
by intros a1 a2 Ha x n'; split; intros [a' ?]; exists a'; simpl; first
[rewrite <-(dist_le _ _ _ _ Ha) by lia|rewrite (dist_le _ _ _ _ Ha) by lia].
Qed.
Global Instance iprop_own_proper :
Proper (() ==> ()) (@iprop_own A) := ne_proper _.
Proper (() ==> ()) (@iprop_own M) := ne_proper _.
(** Introduction and elimination rules *)
Lemma iprop_True_intro P : P True%I.
......@@ -300,14 +300,14 @@ Proof.
Qed.
Lemma iprop_impl_elim P Q : ((P Q) P)%I Q.
Proof. by intros x n ? [HQ HP]; apply HQ. Qed.
Lemma iprop_forall_intro P `(Q: B iProp A): ( b, P Q b) P ( b, Q b)%I.
Proof. by intros HPQ x n ?? b; apply HPQ. Qed.
Lemma iprop_forall_elim `(P : B iProp A) b : ( b, P b)%I P b.
Lemma iprop_forall_intro P `(Q: A iProp M): ( a, P Q a) P ( a, Q a)%I.
Proof. by intros HPQ x n ?? a; apply HPQ. Qed.
Lemma iprop_forall_elim `(P : A iProp M) a : ( a, P a)%I P a.
Proof. intros x n ? HP; apply HP. Qed.
Lemma iprop_exist_intro `(P : B iProp A) b : P b ( b, P b)%I.
Proof. by intros x n ??; exists b. Qed.
Lemma iprop_exist_elim `(P : B iProp A) Q : ( b, P b Q) ( b, P b)%I Q.
Proof. by intros HPQ x n ? [b ?]; apply HPQ with b. Qed.
Lemma iprop_exist_intro `(P : A iProp M) a : P a ( a, P a)%I.
Proof. by intros x n ??; exists a. Qed.
Lemma iprop_exist_elim `(P : A iProp M) Q : ( a, P a Q) ( a, P a)%I Q.
Proof. by intros HPQ x n ? [a ?]; apply HPQ with a. Qed.
(* BI connectives *)
Lemma iprop_sep_elim_l P Q : (P Q)%I P.
......@@ -315,19 +315,19 @@ Proof.
intros x n Hvalid (x1&x2&Hx&?&?); rewrite Hx in Hvalid |- *.
by apply iprop_weaken with x1 n; auto using ra_included_l.
Qed.
Global Instance iprop_sep_left_id : LeftId () True%I (@iprop_sep A).
Global Instance iprop_sep_left_id : LeftId () True%I (@iprop_sep M).
Proof.
intros P x n Hvalid; split.
* intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
apply iprop_weaken with x2 n; auto using ra_included_r.
* by intros ?; exists (unit x), x; rewrite ra_unit_l.
Qed.
Global Instance iprop_sep_commutative : Commutative () (@iprop_sep A).
Global Instance iprop_sep_commutative : Commutative () (@iprop_sep M).
Proof.
by intros P Q x n ?; split;
intros (x1&x2&?&?&?); exists x2, x1; rewrite (commutative op).
Qed.
Global Instance iprop_sep_associative : Associative () (@iprop_sep A).
Global Instance iprop_sep_associative : Associative () (@iprop_sep M).
Proof.
intros P Q R x n ?; split.
* intros (x1&x2&Hx&?&y1&y2&Hy&?&?); exists (x1 y1), y2; split_ands; auto.
......@@ -355,15 +355,15 @@ Proof.
Qed.
Lemma iprop_sep_and P Q R : ((P Q) R)%I ((P R) (Q R))%I.
Proof. by intros x n ? (x1&x2&Hx&[??]&?); split; exists x1, x2. Qed.
Lemma iprop_sep_exist {B} (P : B iProp A) Q :
Lemma iprop_sep_exist `(P : A iProp M) Q :
(( b, P b) Q)%I ( b, P b Q)%I.
Proof.
split; [by intros (x1&x2&Hx&[b ?]&?); exists b, x1, x2|].
intros [b (x1&x2&Hx&?&?)]; exists x1, x2; split_ands; by try exists b.
split; [by intros (x1&x2&Hx&[a ?]&?); exists a, x1, x2|].
intros [a (x1&x2&Hx&?&?)]; exists x1, x2; split_ands; by try exists a.
Qed.
Lemma iprop_sep_forall `(P : B iProp A) Q :
(( b, P b) Q)%I ( b, P b Q)%I.
Proof. by intros x n ? (x1&x2&Hx&?&?); intros b; exists x1, x2. Qed.
Lemma iprop_sep_forall `(P : A iProp M) Q :
(( a, P a) Q)%I ( a, P a Q)%I.
Proof. by intros x n ? (x1&x2&Hx&?&?); intros a; exists x1, x2. Qed.
(* Later *)
Lemma iprop_later_weaken P : P ( P)%I.
......@@ -385,15 +385,15 @@ Lemma iprop_later_and P Q : (▷ (P ∧ Q))%I ≡ (▷ P ∧ ▷ Q)%I.
Proof. by intros x [|n]; split. Qed.
Lemma iprop_later_or P Q : ( (P Q))%I ( P Q)%I.
Proof. intros x [|n]; simpl; tauto. Qed.
Lemma iprop_later_forall `(P : B iProp A) : ( b, P b)%I ( b, P b)%I.
Lemma iprop_later_forall `(P : A iProp M) : ( a, P a)%I ( a, P a)%I.
Proof. by intros x [|n]. Qed.
Lemma iprop_later_exist `(P : B iProp A) : ( b, P b)%I ( b, P b)%I.
Lemma iprop_later_exist `(P : A iProp M) : ( a, P a)%I ( a, P a)%I.
Proof. by intros x [|n]. Qed.
Lemma iprop_later_exist' `{Inhabited B} (P : B iProp A) :
( b, P b)%I ( b, P b)%I.
Lemma iprop_later_exist' `{Inhabited A} (P : A iProp M) :
( a, P a)%I ( a, P a)%I.
Proof.
intros x [|n]; split; try done.
by destruct (_ : Inhabited B) as [b]; exists b.
by destruct (_ : Inhabited A) as [a]; exists a.
Qed.
Lemma iprop_later_sep P Q : ( (P Q))%I ( P Q)%I.
Proof.
......@@ -403,7 +403,7 @@ Proof.
as ([y1 y2]&Hx'&Hy1&Hy2); auto using cmra_valid_S; simpl in *.
exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1, Hy2].
* destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)].
exists x1, x2; eauto using (dist_S (A:=A)).
exists x1, x2; eauto using (dist_S (A:=M)).
Qed.
(* Always *)
......@@ -425,9 +425,9 @@ Lemma iprop_always_and P Q : (□ (P ∧ Q))%I ≡ (□ P ∧ □ Q)%I.
Proof. done. Qed.
Lemma iprop_always_or P Q : ( (P Q))%I ( P Q)%I.
Proof. done. Qed.
Lemma iprop_always_forall `(P : B iProp A) : ( b, P b)%I ( b, P b)%I.
Lemma iprop_always_forall `(P : A iProp M) : ( a, P a)%I ( a, P a)%I.
Proof. done. Qed.
Lemma iprop_always_exist `(P : B iProp A) : ( b, P b)%I ( b, P b)%I.
Lemma iprop_always_exist `(P : A iProp M) : ( a, P a)%I ( a, P a)%I.
Proof. done. Qed.
Lemma iprop_always_and_always_box P Q : ( P Q)%I ( P Q)%I.
Proof.
......@@ -436,7 +436,7 @@ Proof.
Qed.
(* Own *)
Lemma iprop_own_op (a1 a2 : A) :
Lemma iprop_own_op (a1 a2 : M) :
iprop_own (a1 a2) (iprop_own a1 iprop_own a2)%I.
Proof.
intros x n ?; split.
......@@ -446,13 +446,13 @@ Proof.
rewrite (associative op), <-(commutative op z1), <-!(associative op), <-Hy2.
by rewrite (associative op), (commutative op z1), <-Hy1.
Qed.
Lemma iprop_own_valid (a : A) : iprop_own a iprop_valid a.
Lemma iprop_own_valid (a : M) : iprop_own a iprop_valid a.
Proof.
intros x n Hv [a' Hx]; simpl; rewrite Hx in Hv; eauto using cmra_valid_op_l.
Qed.
(* Fix *)
Lemma iprop_fixpoint_unfold (P : iProp A iProp A) `{!Contractive P} :
Lemma iprop_fixpoint_unfold (P : iProp M iProp M) `{!Contractive P} :
iprop_fixpoint P P (iprop_fixpoint P).
Proof. apply fixpoint_unfold. Qed.
End logic.
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