Commit 3ecaaf9b authored by Robbert Krebbers's avatar Robbert Krebbers
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Change UPred to be always valid at step 0.

parent 8b4c7038
......@@ -6,35 +6,39 @@ Local Hint Extern 10 (_ ≤ _) => omega.
Structure uPred (M : cmraT) : Type := IProp {
uPred_holds :> nat M Prop;
uPred_ne x1 x2 n : uPred_holds n x1 x1 ={n}= x2 uPred_holds n x2;
uPred_0 x : uPred_holds 0 x;
uPred_weaken x1 x2 n1 n2 :
x1 x2 n2 n1 validN n2 x2 uPred_holds n1 x1 uPred_holds n2 x2
}.
Arguments uPred_holds {_} _ _ _.
Hint Resolve uPred_0.
Add Printing Constructor uPred.
Instance: Params (@uPred_holds) 3.
Instance uPred_equiv (M : cmraT) : Equiv (uPred M) := λ P Q, x n,
validN n x P n x Q n x.
Instance uPred_dist (M : cmraT) : Dist (uPred M) := λ n P Q, x n',
n' < n validN n' x P n' x Q n' x.
n' n validN n' x P n' x Q n' x.
Program Instance uPred_compl (M : cmraT) : Compl (uPred M) := λ c,
{| uPred_holds n x := c (S n) n x |}.
{| uPred_holds n x := c n n x |}.
Next Obligation. by intros M c x y n ??; simpl in *; apply uPred_ne with x. Qed.
Next Obligation. by intros M c x; simpl. Qed.
Next Obligation.
intros M c x1 x2 n1 n2 ????; simpl in *.
apply (chain_cauchy c (S n2) (S n1)); eauto using uPred_weaken, cmra_valid_le.
apply (chain_cauchy c n2 n1); eauto using uPred_weaken.
Qed.
Instance uPred_cofe (M : cmraT) : Cofe (uPred M).
Proof.
split.
* intros P Q; split; [by intros HPQ n x i ??; apply HPQ|].
intros HPQ x n ?; apply HPQ with (S n); auto.
intros HPQ x n ?; apply HPQ with n; auto.
* intros n; split.
+ by intros P x i.
+ by intros P Q HPQ x i ??; symmetry; apply HPQ.
+ by intros P Q Q' HP HQ x i ??; transitivity (Q i x); [apply HP|apply HQ].
* intros n P Q HPQ x i ??; apply HPQ; auto.
* intros P Q x i ??; lia.
* intros c n x i ??; apply (chain_cauchy c (S i) n); auto.
* 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).
Qed.
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.
......@@ -47,6 +51,7 @@ Program Definition uPred_map {M1 M2 : cmraT} (f : M2 → M1)
`{! n, Proper (dist n ==> dist n) f, !CMRAPreserving 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; simpl; rewrite <-Hy. Qed.
Next Obligation. intros M1 M2 f _ _ P x; apply uPred_0. Qed.
Next Obligation.
by intros M1 M2 f ?? P y1 y2 n i ???; simpl; apply uPred_weaken; auto;
apply validN_preserving || apply included_preserving.
......@@ -72,8 +77,9 @@ Instance uPred_entails {M} : SubsetEq (uPred M) := λ P Q, ∀ x n,
(** logical connectives *)
Program Definition uPred_const {M} (P : Prop) : uPred M :=
{| uPred_holds n x := P |}.
{| uPred_holds n x := match n return _ with 0 => True | _ => P end |}.
Solve Obligations with done.
Next Obligation. intros M P x1 x2 [|n1] [|n2]; auto with lia. Qed.
Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_const True).
Program Definition uPred_and {M} (P Q : uPred M) : uPred M :=
......@@ -92,14 +98,20 @@ Next Obligation.
assert (validN n2 x2') by (by rewrite Hx2'); rewrite <-Hx2'.
by apply HPQ, uPred_weaken with x2' n2, uPred_ne with x2.
Qed.
Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed.
Next Obligation. naive_solver eauto 2 with lia. Qed.
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 :=
{| uPred_holds n x := a, P a n x |}.
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
{| 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.
Program Definition uPred_eq {M} {A : cofeT} (a1 a2 : A) : uPred M :=
{| uPred_holds n x := a1 ={n}= a2 |}.
......@@ -110,6 +122,7 @@ Program Definition uPred_sep {M} (P Q : uPred M) : uPred M :=
Next Obligation.
by intros M P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy.
Qed.
Next Obligation. by intros M P Q x; exists x, x. Qed.
Next Obligation.
intros M P Q x y n1 n2 Hxy ?? (x1&x2&Hx&?&?).
assert ( x2', y ={n2}= x1 x2' x2 x2') as (x2'&Hy&?).
......@@ -131,6 +144,7 @@ Next Obligation.
rewrite <-(dist_le _ _ _ _ Hx) by done; apply HPQ; auto.
by rewrite (dist_le _ _ _ n2 Hx).
Qed.
Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed.
Next Obligation.
intros M P Q x1 x2 n1 n2 ??? HPQ x3 n3 ???; simpl in *.
apply uPred_weaken with (x1 x3) n3; auto using ra_preserving_r.
......@@ -141,6 +155,7 @@ Qed.
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.
Next Obligation. done. Qed.
Next Obligation.
intros M P x1 x2 [|n1] [|n2] ????; auto with lia.
apply uPred_weaken with x1 n1; eauto using cmra_valid_S.
......@@ -148,6 +163,7 @@ Qed.
Program Definition uPred_always {M} (P : uPred M) : uPred M :=
{| uPred_holds n x := P n (unit x) |}.
Next Obligation. by intros M P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed.
Next Obligation. by intros; simpl. Qed.
Next Obligation.
intros M P x1 x2 n1 n2 ????; eapply uPred_weaken with (unit x1) n1;
auto using ra_unit_preserving, cmra_unit_valid.
......@@ -156,13 +172,14 @@ Qed.
Program Definition uPred_own {M : cmraT} (a : M) : uPred M :=
{| uPred_holds n x := a', x ={n}= a a' |}.
Next Obligation. by intros M a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed.
Next Obligation. by intros M a x; exists x. Qed.
Next Obligation.
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 uPred_valid {M : cmraT} (a : M) : uPred M :=
{| uPred_holds n x := validN n a |}.
Solve Obligations with naive_solver eauto 2 using cmra_valid_le.
Solve Obligations with naive_solver eauto 2 using cmra_valid_le, cmra_valid_0.
Delimit Scope uPred_scope with I.
Bind Scope uPred_scope with uPred.
......@@ -205,7 +222,7 @@ Qed.
(** Non-expansiveness *)
Global Instance uPred_const_proper : Proper (iff ==> ()) (@uPred_const M).
Proof. intros P Q HPQ ???; apply HPQ. Qed.
Proof. by intros P Q HPQ ? [|n] ?; try apply HPQ. Qed.
Global Instance uPred_and_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
Proof.
......@@ -263,12 +280,12 @@ Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
Global Instance uPred_exists_ne {A : cofeT} :
Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
Proof.
by intros n P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
by intros n P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
Qed.
Global Instance uPred_exist_proper {A : cofeT} :
Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
Proof.
by intros P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12.
by intros P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
Qed.
Global Instance uPred_later_contractive : Contractive (@uPred_later M).
Proof.
......@@ -291,11 +308,11 @@ Global Instance uPred_own_proper :
(** Introduction and elimination rules *)
Lemma uPred_const_intro P (Q : Prop) : Q P uPred_const Q.
Proof. by intros ???. Qed.
Proof. by intros ?? [|?]. Qed.
Lemma uPred_True_intro P : P True%I.
Proof. done. Qed.
Proof. by apply uPred_const_intro. Qed.
Lemma uPred_False_elim P : False%I P.
Proof. by intros x n ?. Qed.
Proof. by intros x [|n] ?. Qed.
Lemma uPred_and_elim_l P Q : (P Q)%I P.
Proof. by intros x n ? [??]. Qed.
Lemma uPred_and_elim_r P Q : (P Q)%I Q.
......@@ -319,9 +336,9 @@ Proof. by intros HPQ x n ?? a; apply HPQ. Qed.
Lemma uPred_forall_elim `(P : A uPred M) a : ( a, P a)%I P a.
Proof. intros x n ? HP; apply HP. Qed.
Lemma uPred_exist_intro `(P : A uPred M) a : P a ( a, P a)%I.
Proof. by intros x n ??; exists a. Qed.
Proof. by intros x [|n] ??; [done|exists a]. Qed.
Lemma uPred_exist_elim `(P : A uPred M) Q : ( a, P a Q) ( a, P a)%I Q.
Proof. by intros HPQ x n ? [a ?]; apply HPQ with a. Qed.
Proof. by intros HPQ x [|n] ?; [|intros [a ?]; apply HPQ with a]. Qed.
(* BI connectives *)
Lemma uPred_sep_elim_l P Q : (P Q)%I P.
......@@ -334,7 +351,7 @@ Proof.
intros P x n Hvalid; split.
* intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
apply uPred_weaken with x2 n; auto using ra_included_r.
* by intros ?; exists (unit x), x; rewrite ra_unit_l.
* by destruct n as [|n]; [|intros ?; exists (unit x), x; rewrite ra_unit_l].
Qed.
Global Instance uPred_sep_commutative : Commutative () (@uPred_sep M).
Proof.
......@@ -372,7 +389,7 @@ Proof. by intros x n ? (x1&x2&Hx&[??]&?); split; exists x1, x2. Qed.
Lemma uPred_sep_exist `(P : A uPred M) Q :
(( b, P b) Q)%I ( b, P b Q)%I.
Proof.
split; [by intros (x1&x2&Hx&[a ?]&?); exists a, x1, x2|].
intros x [|n]; [done|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 uPred_sep_forall `(P : A uPred M) Q :
......@@ -402,13 +419,10 @@ Proof. intros x [|n]; simpl; tauto. Qed.
Lemma uPred_later_forall `(P : A uPred M) : ( a, P a)%I ( a, P a)%I.
Proof. by intros x [|n]. Qed.
Lemma uPred_later_exist `(P : A uPred M) : ( a, P a)%I ( a, P a)%I.
Proof. by intros x [|n]. Qed.
Proof. by intros x [|[|n]]. Qed.
Lemma uPred_later_exist' `{Inhabited A} (P : A uPred M) :
( a, P a)%I ( a, P a)%I.
Proof.
intros x [|n]; split; try done.
by destruct (_ : Inhabited A) as [a]; exists a.
Qed.
Proof. intros x [|[|n]]; split; done || by exists inhabitant; simpl. Qed.
Lemma uPred_later_sep P Q : ( (P Q))%I ( P Q)%I.
Proof.
intros x n ?; split.
......
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