Commit 4e756c6d authored by Robbert Krebbers's avatar Robbert Krebbers

Finite powerset RA.

parent 4d5474e2
From iris.algebra Require Export gmap.
From iris.algebra Require Import excl.
From iris.prelude Require Import mapset.
Definition gsetC K `{Countable K} := gmapC K (exclC unitC).
Definition to_gsetC `{Countable K} (X : gset K) : gsetC K :=
to_gmap (Excl ()) X.
Section gset.
Context `{Countable K}.
Implicit Types X Y : gset K.
Lemma to_gsetC_empty : to_gsetC ( : gset K) = .
Proof. apply to_gmap_empty. Qed.
Lemma to_gsetC_union X Y : X Y to_gsetC X to_gsetC Y = to_gsetC (X Y).
Proof.
intros HXY; apply: map_eq=> i; rewrite /to_gsetC /=.
rewrite lookup_op !lookup_to_gmap. repeat case_option_guard; set_solver.
Qed.
Lemma to_gsetC_valid X : to_gsetC X.
Proof. intros i. rewrite /to_gsetC lookup_to_gmap. by case_option_guard. Qed.
Lemma to_gsetC_valid_op X Y : (to_gsetC X to_gsetC Y) X Y.
Proof.
split; last (intros; rewrite to_gsetC_union //; apply to_gsetC_valid).
intros HXY i ??; move: (HXY i); rewrite lookup_op !lookup_to_gmap.
rewrite !option_guard_True //.
Qed.
Context `{Fresh K (gset K), !FreshSpec K (gset K)}.
Lemma updateP_alloc_strong (Q : gsetC K Prop) (I : gset K) X :
( i, i X i I Q (to_gsetC ({[i]} X))) to_gsetC X ~~>: Q.
Proof.
intros; apply updateP_alloc_strong with I (Excl ()); [done|]=> i.
rewrite /to_gsetC lookup_to_gmap_None -to_gmap_union_singleton; eauto.
Qed.
Lemma updateP_alloc (Q : gsetC K Prop) X :
( i, i X Q (to_gsetC ({[i]} X))) to_gsetC X ~~>: Q.
Proof. move=>??. eapply updateP_alloc_strong with (I:=); by eauto. Qed.
Lemma updateP_alloc_strong' (I : gset K) X :
to_gsetC X ~~>: λ Y : gsetC K, i, Y = to_gsetC ({[ i ]} X) i I i X.
Proof. eauto using updateP_alloc_strong. Qed.
Lemma updateP_alloc' X :
to_gsetC X ~~>: λ Y : gsetC K, i, Y = to_gsetC ({[ i ]} X) i X.
Proof. eauto using updateP_alloc. Qed.
End gset.
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