Commit de261ce8 authored by Ralf Jung's avatar Ralf Jung

characterize inclusion of sts_frag

I thought I could derive an interesting Lemma from this. Turns out I cannot. But this one may still be useful, at some point.
parent ba30e046
......@@ -103,15 +103,21 @@ Proof.
Qed.
Instance up_proper : Proper ((=) ==> () ==> ()) up.
Proof. by intros ??? ?? [??]; split; apply up_preserving. Qed.
Instance up_set_preserving : Proper (() ==> flip () ==> ()) up_set.
Proof.
intros S1 S2 HS T1 T2 HT. rewrite /up_set.
f_equiv; last done. move =>s1 s2 Hs. simpl in HT. by apply up_preserving.
Qed.
Instance up_set_proper : Proper (() ==> () ==> ()) up_set.
Proof.
intros S1 S2 HS T1 T2 HT. rewrite /up_set HS.
f_equiv=>s1 s2 Hs. by rewrite Hs HT.
by intros ?? EQ1 ?? EQ2; split; apply up_set_preserving; rewrite ?EQ1 ?EQ2.
Qed.
Lemma elem_of_up s T : s up s T.
Proof. constructor. Qed.
Lemma subseteq_up_set S T : S up_set S T.
Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed.
Lemma up_up_set s T : up s T up_set {[ s ]} T.
Proof. by rewrite /up_set collection_bind_singleton. Qed.
Lemma closed_up_set S T :
( s, s S tok s T ) S closed (up_set S T) T.
Proof.
......@@ -213,6 +219,7 @@ Context {A B : Type} (R : relation A) (tok : A → set B).
Canonical Structure stsRA := validityRA (sts R tok).
Definition sts_auth (s : A) (T : set B) : stsRA := to_validity (auth s T).
Definition sts_frag (S : set A) (T : set B) : stsRA := to_validity (frag S T).
Lemma sts_update s1 s2 T1 T2 :
sts.step R tok (s1,T1) (s2,T2) sts_auth s1 T1 ~~> sts_auth s2 T2.
Proof.
......@@ -220,4 +227,24 @@ Proof.
destruct (sts.step_closed R tok s1 s2 T1 T2 S Tf) as (?&?&?); auto.
repeat (done || constructor).
Qed.
Lemma sts_frag_included S1 S2 T1 T2 Tdf :
sts.closed R tok S1 T1 sts.closed R tok S2 T2
T2 T1 Tdf T1 Tdf
S2 (S1 sts.up_set R tok S2 Tdf)
sts_frag S1 T1 sts_frag S2 T2.
Proof.
move=>Hcl1 Hcl2 Htk Hdf Hs. exists (sts_frag (sts.up_set R tok S2 Tdf) Tdf).
split; first split; simpl.
- intros _. split_ands.
+ done.
+ apply sts.closed_up_set.
* move=>s Hs2. move:(sts.closed_disjoint _ _ _ _ Hcl2 _ Hs2).
solve_elem_of +Htk.
* by eapply sts.closed_ne.
+ constructor; last done. rewrite -Hs. by eapply sts.closed_ne.
- done.
- intros _. constructor; [ solve_elem_of +Htk | done].
Qed.
End stsRA.
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