gmultiset RA

_CoqProject

@@ -22,6 +22,7 @@ theories/algebra/vector.v
 theories/algebra/updates.v
 theories/algebra/local_updates.v
 theories/algebra/gset.v
+theories/algebra/gmultiset.v
 theories/algebra/coPset.v
 theories/algebra/deprecated.v
 theories/base_logic/upred.v

theories/algebra/gmultiset.v

+From iris.algebra Require Export cmra.
+From iris.algebra Require Import updates local_updates.
+From stdpp Require Export collections gmultiset countable.
+Set Default Proof Using "Type".
+
+(* The multiset union CMRA *)
+Section gmultiset.
+  Context `{Countable K}.
+  Implicit Types X Y : gmultiset K.
+
+  Canonical Structure gmultisetC := discreteC (gmultiset K).
+
+  Instance gmultiset_valid : Valid (gmultiset K) := λ _, True.
+  Instance gmultiset_validN : ValidN (gmultiset K) := λ _ _, True.
+  Instance gmultiset_unit : Unit (gmultiset K) := (∅ : gmultiset K).
+  Instance gmultiset_op : Op (gmultiset K) := union.
+  Instance gmultiset_pcore : PCore (gmultiset K) := λ X, Some ∅.
+
+  Lemma gmultiset_op_union X Y : X ⋅ Y = X ∪ Y.
+  Proof. done. Qed.
+  Lemma gmultiset_core_empty X : core X = ∅.
+  Proof. done. Qed.
+  Lemma gmultiset_included X Y : X ≼ Y ↔ X ⊆ Y.
+  Proof.
+    split.
+    - intros [Z ->%leibniz_equiv].
+      rewrite gmultiset_op_union. apply gmultiset_union_subseteq_l.
+    - intros ->%gmultiset_union_difference. by exists (Y ∖ X).
+  Qed.
+
+  Lemma gmultiset_ra_mixin : RAMixin (gmultiset K).
+  Proof.
+    apply ra_total_mixin; eauto.
+    - by intros X Y Z ->%leibniz_equiv.
+    - by intros X Y ->%leibniz_equiv.
+    - solve_proper.
+    - intros X1 X2 X3. by rewrite !gmultiset_op_union assoc_L.
+    - intros X1 X2. by rewrite !gmultiset_op_union comm_L.
+    - intros X. by rewrite gmultiset_core_empty left_id.
+    - intros X1 X2 HX. rewrite !gmultiset_core_empty. exists ∅.
+      by rewrite left_id.
+  Qed.
+
+  Canonical Structure gmultisetR := discreteR (gmultiset K) gmultiset_ra_mixin.
+
+  Global Instance gmultiset_cmra_discrete : CmraDiscrete gmultisetR.
+  Proof. apply discrete_cmra_discrete. Qed.
+
+  Lemma gmultiset_ucmra_mixin : UcmraMixin (gmultiset K).
+  Proof. split. done. intros X. by rewrite gmultiset_op_union left_id_L. done. Qed.
+  Canonical Structure gmultisetUR := UcmraT (gmultiset K) gmultiset_ucmra_mixin.
+
+  Lemma gmultiset_opM X mY : X ⋅? mY = X ∪ from_option id ∅ mY.
+  Proof. destruct mY; by rewrite /= ?right_id_L. Qed.
+
+  Lemma gmultiset_update X Y : X ~~> Y.
+  Proof. done. Qed.
+
+  Lemma gmultiset_local_update_alloc X Y X' : (X,Y) ~l~> (X ∪ X', Y ∪ X').
+  Proof.
+    rewrite local_update_unital_discrete=> Z' _ /leibniz_equiv_iff->.
+    split. done. rewrite !gmultiset_op_union.
+    by rewrite -!assoc (comm _ Z' X').
+  Qed.
+
+  Lemma gmultiset_local_update_dealloc X Y X' : X' ⊆ X → X' ⊆ Y → (X,Y) ~l~> (X ∖ X', Y ∖ X').
+  Proof.
+    intros ->%gmultiset_union_difference ->%gmultiset_union_difference.
+    rewrite local_update_unital_discrete=> Z' _ /leibniz_equiv_iff->.
+    split. done. rewrite !gmultiset_op_union=> x.
+    repeat (rewrite multiplicity_difference || rewrite multiplicity_union).
+    omega.
+  Qed.
+
+End gmultiset.
+
+Arguments gmultisetC _ {_ _}.
+Arguments gmultisetR _ {_ _}.
+Arguments gmultisetUR _ {_ _}.