Commit 9f8d9a5e authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan

Prove that uPred is complete even if we remove the validity

restriction in uPred_closed.
parent 450aef18
......@@ -194,7 +194,6 @@ Lemma nnupd_nnupd_k_dist k P: (|=n=> P)%I ≡{k}≡ (|=n=>_k P)%I.
eapply laterN_big; eauto. unseal.
eapply (Hnnupdk n k); first omega; eauto.
exists x, x'. split_and!; eauto. eapply uPred_closed; eauto.
eapply cmra_validN_op_l; eauto.
** intros HP. eapply IHk; auto.
move:HP. unseal. intros (?&?); naive_solver.
Qed.
......@@ -234,7 +233,7 @@ Proof.
apply and_intro.
* rewrite (nnupd_k_unfold k P). rewrite and_elim_l.
rewrite nnupd_k_unfold. rewrite and_elim_l.
apply wand_intro_l.
apply wand_intro_l.
rewrite {1}(nnupd_k_intro (S k) (P - ^(S k) (False)%I)).
rewrite nnupd_k_unfold and_elim_l. apply wand_elim_r.
* do 2 rewrite nnupd_k_weaken //.
......
......@@ -76,7 +76,7 @@ Next Obligation.
by rewrite Hy Hx assoc.
Qed.
Next Obligation.
intros M P Q n1 n2 x (x1&x2&Hx&?&?) ?; rewrite {1}(dist_le _ _ _ _ Hx) // =>?.
intros M P Q n1 n2 x (x1&x2&Hx&?&?) ?.
exists x1, x2; ofe_subst; split_and!;
eauto using dist_le, uPred_closed, cmra_validN_op_l, cmra_validN_op_r.
Qed.
......
......@@ -14,35 +14,7 @@ Record uPred (M : ucmraT) : Type := IProp {
it to only valid elements. *)
uPred_mono n x1 x2 : uPred_holds n x1 x1 {n} x2 uPred_holds n x2;
(* We have to restrict this to hold only for valid elements,
otherwise this condition is no longer limit preserving, and uPred
does no longer form a COFE (i.e., [uPred_compl] breaks). This is
because the distance and equivalence on this cofe ignores the
truth value on invalid elements. This, in turn, is required by
the fact that entailment has to ignore invalid elements, which is
itself essential for proving [ownM_valid].
We could, actually, remove this restriction and make this
condition apply even to invalid elements: we have proved that
uPred is isomorphic to a sub-COFE of the COFE of predicates that
are monotonous both with respect to the step index and with
respect to x. However, that would essentially require changing
(by making it more complicated) the model of many connectives of
the logic, which we don't want.
This sub-COFE is the sub-COFE of monotonous sProp predicates P
such that the following sProp assertion is valid:
∀ x, (V(x) → P(x)) → P(x)
Where V is the validity predicate.
Another way of saying that this is equivalent to this definition of
uPred is to notice that our definition of uPred is equivalent to
quotienting the COFE of monotonous sProp predicates with the
following (sProp) equivalence relation:
P1 ≡ P2 := ∀ x, V(x) → (P1(x) ↔ P2(x))
whose equivalence classes appear to all have one only canonical
representative such that ∀ x, (V(x) → P(x)) → P(x).
*)
uPred_closed n1 n2 x : uPred_holds n1 x n2 n1 {n2} x uPred_holds n2 x
uPred_closed n1 n2 x : uPred_holds n1 x n2 n1 uPred_holds n2 x
}.
Arguments uPred_holds {_} _ _ _ : simpl never.
Add Printing Constructor uPred.
......@@ -55,6 +27,39 @@ Arguments uPred_holds {_} _%I _ _.
Section cofe.
Context {M : ucmraT}.
(* A good way of understanding this defintion of the uPred OFE is to
consider the OFE uPred0 of monotonous SProp predicates. That is,
uPred0 is the OFE of non-expanding functions from M to SProp that
are monotonous with respect to CMRA inclusion. This notion of
monotonicity has to be stated in the SProp logic. It is exactly
uPred_mono.
Then, we quotient uPred0 *in the sProp logic* with respect to
equivalence on valid elements of M. That is, we quotient with
respect to the following *sProp* equivalence relation:
P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x)) (1)
When seen from the ambiant logic, computing this logic require
redefinig both a custom Equiv and Dist.
It is worth noting that this equivalence relation admit canonical
representatives. More precisely, one can show that every
equivalence class contain exactly one element P0 such that:
∀ x, (✓ x → P(x)) → P(x) (2)
(Again, this assertion has to be understood in sProp). Starting
from an element P of a given class, one can build this canonical
representative by chosing:
P0(x) = ✓ x → P(x) (3)
Hence, as an alternative definition of uPred, we could use the set
of canonical representatives (i.e., the subtype of monotonous
sProp predicates that verify (2)). This alternative definition would
save us from using a quotient. However, the definitions of the various
connectives would get more complicated, because we have to make sure
they all verify (2), which sometimes requires some adjustments. We would
moreover need to prove one more property for every logical connective.
*)
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'.
......@@ -77,15 +82,20 @@ Section cofe.
Canonical Structure uPredC : ofeT := OfeT (uPred M) uPred_ofe_mixin.
Program Definition uPred_compl : Compl uPredC := λ c,
{| uPred_holds n x := c n n x |}.
Next Obligation. naive_solver eauto using uPred_mono. Qed.
{| uPred_holds n x := n', n' n {n'}x c n' n' x |}.
Next Obligation.
move=> /= c n x1 x2 Hx1 Hx12 n' Hn'n Hv.
eapply uPred_mono. by eapply Hx1, cmra_validN_includedN, cmra_includedN_le.
by eapply cmra_includedN_le.
Qed.
Next Obligation.
intros c n1 n2 x ???; simpl in *.
apply (chain_cauchy c n2 n1); eauto using uPred_closed.
move=> /= c n1 n2 x Hc Hn12 n3 Hn23 Hv. apply Hc. lia. done.
Qed.
Global Program Instance uPred_cofe : Cofe uPredC := {| compl := uPred_compl |}.
Next Obligation.
intros n c; split=>i x ??; symmetry; apply (chain_cauchy c i n); auto.
intros n c; split=>i x Hin Hv.
etrans; [|by symmetry; apply (chain_cauchy c i n)]. split=>H; [by apply H|].
repeat intro. apply (chain_cauchy c n' i)=>//. by eapply uPred_closed.
Qed.
End cofe.
Arguments uPredC : clear implicits.
......
From iris.base_logic Require Export upred.
Record sProp := SProp {
sProp_holds :> nat Prop;
sProp_holds_mono : Proper (flip () ==> impl) sProp_holds
}.
Existing Instance sProp_holds_mono.
Instance sprop_holds_mono_flip (P : sProp) : Proper (() ==> flip impl) P.
Proof. solve_proper. Qed.
Arguments sProp_holds _ _ : simpl never.
Add Printing Constructor sProp.
Section sProp_cofe.
Inductive sProp_equiv' (P Q : sProp) : Prop :=
{ sProp_in_equiv : n, P n Q n }.
Instance sProp_equiv : Equiv sProp := sProp_equiv'.
Inductive sProp_dist' (n : nat) (P Q : sProp) : Prop :=
{ sProp_in_dist : n', n' n P n' Q n' }.
Instance sProp_dist : Dist sProp := sProp_dist'.
Definition sProp_ofe_mixin : OfeMixin sProp.
Proof.
split.
- intros P Q; split.
+ intros HPQ n; split=>??; apply HPQ.
+ intros HPQ; split=>?; eapply HPQ; auto.
- intros n; split.
+ by intros P; split.
+ by intros P Q HPQ; split=> ??; symmetry; apply HPQ.
+ by intros P Q Q' HP HQ; split=> n' ?; trans (Q n'); [apply HP|apply HQ].
- intros n P Q HPQ; split=> ??. apply HPQ. auto.
Qed.
Canonical Structure sPropC : ofeT := OfeT (sProp) sProp_ofe_mixin.
Program Definition sProp_compl : Compl sPropC := λ c,
{| sProp_holds n := c n n |}.
Next Obligation.
move => c n1 n2 Hn12 /(sProp_holds_mono _ _ _ Hn12) H.
by apply (chain_cauchy c _ _ Hn12).
Qed.
Global Program Instance sProp_cofe : Cofe sPropC := {| compl := sProp_compl |}.
Next Obligation. intros n. split=>n' Hn'. symmetry. by apply (chain_cauchy c). Qed.
End sProp_cofe.
Arguments sPropC : clear implicits.
Instance sProp_proper : Proper (() ==> (=) ==> iff) sProp_holds.
Proof. by move=> ??[?]??->. Qed.
Global Instance limit_preserving_sprop `{Cofe A} (P : A sPropC) :
NonExpansive P LimitPreserving (λ x, n, P x n).
Proof.
intros ? c Hc n. specialize (Hc n n).
assert (Hcompl : P (compl c) {n} P (c n)) by by rewrite (conv_compl n c).
by apply Hcompl.
Qed.
Section uPred1.
Context {M : ucmraT}.
(* TODO : use logical connectives in sProp *)
Program Definition uPred1_valid (P : M -n> sPropC) : sPropC :=
SProp (λ n, n', n' n
( x1 x2, x1 {n'} x2 P x1 n' P x2 n')
( x, ( n'', n'' n' {n''} x P x n'') P x n')) _.
Next Obligation. move=>??? /= H. by setoid_rewrite H. Qed.
Instance uPred1_valid_ne : NonExpansive uPred1_valid.
Proof.
split=>??. repeat ((apply forall_proper=>?) || f_equiv).
- split=> HH ?; apply H, HH, H=>//; lia.
- split=> HH ?; (apply H, HH; [lia|])=> ???; (apply H; [lia|auto]).
Qed.
Definition uPred1 :=
sigC (λ (P : M -n> sPropC), n, uPred1_valid P n).
Global Instance uPred1_cofe : Cofe uPred1.
Proof. eapply @sig_cofe, limit_preserving_sprop, _. Qed.
Program Definition uPred_of_uPred1 (P : uPred1) : uPred M :=
{| uPred_holds n x := `P x n |}.
Next Obligation.
intros P ?????. by eapply (proj1 (proj2_sig P _ _ (le_refl _))).
Qed.
Next Obligation. move => /= ????? Hle. by rewrite ->Hle. Qed.
Global Instance uPred_of_uPred1_ne : NonExpansive uPred_of_uPred1.
Proof.
move=>??? EQ. split=>? x ??. specialize (EQ x). by split=>H; apply EQ.
Qed.
Global Instance uPred_of_uPred1_proper : Proper (() ==> ()) uPred_of_uPred1.
Proof. apply (ne_proper _). Qed.
Program Definition uPred1_of_uPred (P : uPred M) : uPred1 :=
λne x, SProp (λ n, n', n' n {n'}x P n' x) _.
Next Obligation. move => ???? /= Hle ?. by setoid_rewrite -> Hle. Qed.
Next Obligation.
move => ???? H. split=>??. do 2 apply forall_proper=>?.
split=>HH ?;
(eapply uPred_ne; [eapply dist_le; [by symmetry|lia]|]);
(eapply HH, cmra_validN_ne; [eapply dist_le|]=>//; lia).
Qed.
Next Obligation.
split.
- move=>??? HP ???. eapply uPred_mono, cmra_includedN_le=>//.
eapply HP, cmra_validN_includedN=>//. eapply cmra_includedN_le=>//.
- move=>? HP ???. by eapply HP.
Qed.
Global Instance uPred1_of_uPred_ne : NonExpansive uPred1_of_uPred.
Proof.
move=>??? EQ. split=>??. do 3 apply forall_proper=>?. apply EQ=>//. lia.
Qed.
Global Instance uPred1_of_uPred_proper : Proper (() ==> ()) uPred1_of_uPred.
Proof. apply (ne_proper _). Qed.
Lemma uPred1_of_uPred_of_uPred1 P : uPred1_of_uPred (uPred_of_uPred1 P) P.
Proof.
split=>?; split=>HP.
- apply (proj2 (proj2_sig P _ _ (le_refl _)))=>???. by apply HP.
- move=>? Hle ?. unfold uPred_holds=>/=. by rewrite ->Hle.
Qed.
Lemma uPred_of_uPred1_of_uPred P : uPred_of_uPred1 (uPred1_of_uPred P) P.
Proof.
split=>?; split=>HP.
- by apply HP.
- move=>???. by eapply uPred_closed.
Qed.
End uPred1.
Arguments uPred1 _ : clear implicits.
Section uPred2.
Context {M : ucmraT}.
Record uPred2 : Type := IProp {
uPred2_holds :> M sProp;
uPred2_mono n x1 x2 : uPred2_holds x1 n x1 {n} x2 uPred2_holds x2 n
}.
Section cofe.
Inductive uPred2_equiv' (P Q : uPred2) : Prop :=
{ uPred_in_equiv : n x, {n} x P x n Q x n }.
Instance uPred2_equiv : Equiv uPred2 := uPred2_equiv'.
Inductive uPred2_dist' (n : nat) (P Q : uPred2) : Prop :=
{ uPred_in_dist : n' x, n' n {n'} x P x n' Q x n' }.
Instance uPred_dist : Dist uPred2 := uPred2_dist'.
Definition uPred2_ofe_mixin : OfeMixin uPred2.
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 x i);[apply HP|apply HQ].
- intros n P Q HPQ; split=> i x ??; apply HPQ; auto.
Qed.
Canonical Structure uPred2C : ofeT := OfeT uPred2 uPred2_ofe_mixin.
Program Definition uPred2_compl : Compl uPred2C := λ c,
{| uPred2_holds x := SProp (λ n, n', n' n {n'}x c n x n') _ |}.
Next Obligation.
move => c ? n1 n2 /= Hn12 H ???. apply (chain_cauchy c _ _ Hn12)=>//.
apply H. lia. done.
Qed.
Next Obligation.
move => ???? HH ????. eapply uPred2_mono, cmra_includedN_le=>//.
apply HH=>//. eapply cmra_validN_includedN=>//.
by eapply cmra_includedN_le.
Qed.
Global Program Instance uPred2_cofe :
Cofe uPred2C := {| compl := uPred2_compl |}.
Next Obligation.
intro n. split. move => ?? Hle. split=>HP.
- apply (chain_cauchy c _ _ Hle)=>//. by apply HP.
- move=> ???. eapply sProp_holds_mono=>//.
eapply (chain_cauchy c _ n), HP=>//.
Qed.
End cofe.
Program Definition uPred2_of_uPred1 (P : uPred1 M) : uPred2 :=
{| uPred2_holds x := `P x |}.
Next Obligation.
move=>/= P ?????. by eapply (proj1 (proj2_sig P _ _ (le_refl _))).
Qed.
Global Instance uPred2_of_uPred1_ne : NonExpansive uPred2_of_uPred1.
Proof. move=>??? EQ. split=>????. by apply EQ. Qed.
Global Instance uPred2_of_uPred1_proper : Proper (() ==> ()) uPred2_of_uPred1.
Proof. apply (ne_proper _). Qed.
Program Definition uPred1_of_uPred2 (P : uPred2) : uPred1 M :=
λne x, SProp (λ n, n', n' n {n'}x P x n') _.
Next Obligation. move=>???? HLE ??. rewrite <-HLE. auto. Qed.
Next Obligation.
split=>??. do 2 apply forall_proper=>?.
split=>HH ?; (eapply uPred2_mono, cmra_includedN_le;
[eapply HH, cmra_validN_ne; [eapply dist_le; [done|lia]|done]|
by rewrite H|lia]).
Qed.
Next Obligation.
split.
- move=>??? HH ???. eapply uPred2_mono; [|by eapply cmra_includedN_le].
by eapply HH, cmra_validN_includedN, cmra_includedN_le.
- move=>? HH ???. by eapply HH.
Qed.
Global Instance uPred1_of_uPred2_ne : NonExpansive uPred1_of_uPred2.
Proof.
move=> ??? EQ. split=>??. do 3 eapply forall_proper=>?.
apply EQ. lia. done.
Qed.
Lemma uPred1_of_uPred2_of_uPred1 P : uPred1_of_uPred2 (uPred2_of_uPred1 P) P.
Proof.
split=>?. split=>HP.
- apply (proj2 (proj2_sig P _ _ (le_refl _))). intros. by apply HP.
- intros ???. by eapply sProp_holds_mono, HP.
Qed.
Lemma uPred2_of_uPred1_of_uPred2 P : uPred2_of_uPred1 (uPred1_of_uPred2 P) P.
Proof.
split=>?. split=>HP.
- by apply HP.
- intros ???. by eapply sProp_holds_mono, HP.
Qed.
End uPred2.
Arguments uPred2 _ : clear implicits.
Arguments uPred2C _ : clear implicits.
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