more nits

theories/heap_lang/lang.v

@@ -442,10 +442,10 @@ Definition vals_cas_compare_safe (vl v1 : val) : Prop :=
   val_is_unboxed vl ∨ val_is_unboxed v1.
 Arguments vals_cas_compare_safe !_ !_ /.
 
-Definition state_upd_heap (f: gmap loc val → gmap loc val) (σ: state) :=
+Definition state_upd_heap (f: gmap loc val → gmap loc val) (σ: state) : state :=
   {| heap := f σ.(heap); used_proph_id := σ.(used_proph_id) |}.
 Arguments state_upd_heap _ !_ /.
-Definition state_upd_used_proph_id (f: gset proph_id → gset proph_id) (σ: state) :=
+Definition state_upd_used_proph_id (f: gset proph_id → gset proph_id) (σ: state) : state :=
   {| heap := σ.(heap); used_proph_id := f σ.(used_proph_id) |}.
 Arguments state_upd_used_proph_id _ !_ /.
 

theories/heap_lang/proph_map.v

@@ -7,54 +7,6 @@ Import uPred.
 Definition proph_map (P V : Type) `{Countable P} := gmap P (option V).
 Definition proph_val_list (P V : Type) := list (P * V).
 
-Section first_resolve.
-  Context {P V : Type} `{Countable P}.
-  Implicit Type pvs : proph_val_list P V.
-  Implicit Type p : P.
-  Implicit Type v : V.
-  Implicit Type R : proph_map P V.
-
-  (** The first resolve for [p] in [pvs] *)
-  Definition first_resolve pvs p :=
-    (map_of_list pvs : gmap P V) !! p.
-
-  Definition first_resolve_in_list R pvs :=
-    ∀ p v, p ∈ dom (gset _) R →
-           first_resolve pvs p = Some v →
-           R !! p = Some (Some v).
-
-  Lemma first_resolve_insert pvs p R :
-    first_resolve_in_list R pvs →
-    p ∉ dom (gset _) R →
-    first_resolve_in_list (<[p := first_resolve pvs p]> R) pvs.
-  Proof.
-    intros Hf Hnotin p' v' Hp'. rewrite (dom_insert_L R p) in Hp'.
-    erewrite elem_of_union in Hp'. destruct Hp' as [->%elem_of_singleton | Hin] => [->].
-    - by rewrite lookup_insert.
-    - rewrite lookup_insert_ne; first auto. by intros ->.
-  Qed.
-
-  Lemma first_resolve_delete pvs p v R :
-    first_resolve_in_list R ((p, v) :: pvs) →
-    first_resolve_in_list (delete p R) pvs.
-  Proof.
-    intros Hfr p' v' Hpin Heq. rewrite dom_delete_L in Hpin. rewrite /first_resolve in Heq.
-    apply elem_of_difference in Hpin as [Hpin Hne%not_elem_of_singleton].
-    erewrite <- lookup_insert_ne in Heq; last done. rewrite lookup_delete_ne; eauto.
-  Qed.
-
-  Lemma first_resolve_eq R p v w pvs :
-    first_resolve_in_list R ((p, v) :: pvs) →
-    R !! p = Some w →
-    w = Some v.
-  Proof.
-    intros Hfr Hlookup. specialize (Hfr p v (elem_of_dom_2 _ _ _ Hlookup)).
-    rewrite /first_resolve lookup_insert in Hfr. rewrite Hfr in Hlookup; last done.
-    inversion Hlookup. done.
-  Qed.
-
-End first_resolve.
-
 Definition proph_mapUR (P V : Type) `{Countable P} : ucmraT :=
   gmapUR P $ exclR $ optionC $ leibnizC V.
 
@@ -81,6 +33,15 @@ Proof. solve_inG. Qed.
 Section definitions.
   Context `{pG : proph_mapG P V Σ}.
 
+  (** The first resolve for [p] in [pvs] *)
+  Definition first_resolve (pvs : proph_val_list P V) (p : P) :=
+    (map_of_list pvs : gmap P V) !! p.
+
+  Definition first_resolve_in_list (R : proph_map P V) pvs :=
+    ∀ p v, p ∈ dom (gset _) R →
+           first_resolve pvs p = Some v →
+           R !! p = Some (Some v).
+
   Definition proph_map_auth (R : proph_map P V) : iProp Σ :=
     own (proph_map_name pG) (● (to_proph_map R)).
 
@@ -98,6 +59,45 @@ Section definitions.
 
 End definitions.
 
+Section first_resolve.
+  Context {P V : Type} `{Countable P}.
+  Implicit Type pvs : proph_val_list P V.
+  Implicit Type p : P.
+  Implicit Type v : V.
+  Implicit Type R : proph_map P V.
+
+  Lemma first_resolve_insert pvs p R :
+    first_resolve_in_list R pvs →
+    p ∉ dom (gset _) R →
+    first_resolve_in_list (<[p := first_resolve pvs p]> R) pvs.
+  Proof.
+    intros Hf Hnotin p' v' Hp'. rewrite (dom_insert_L R p) in Hp'.
+    erewrite elem_of_union in Hp'. destruct Hp' as [->%elem_of_singleton | Hin] => [->].
+    - by rewrite lookup_insert.
+    - rewrite lookup_insert_ne; first auto. by intros ->.
+  Qed.
+
+  Lemma first_resolve_delete pvs p v R :
+    first_resolve_in_list R ((p, v) :: pvs) →
+    first_resolve_in_list (delete p R) pvs.
+  Proof.
+    intros Hfr p' v' Hpin Heq. rewrite dom_delete_L in Hpin. rewrite /first_resolve in Heq.
+    apply elem_of_difference in Hpin as [Hpin Hne%not_elem_of_singleton].
+    erewrite <- lookup_insert_ne in Heq; last done. rewrite lookup_delete_ne; eauto.
+  Qed.
+
+  Lemma first_resolve_eq R p v w pvs :
+    first_resolve_in_list R ((p, v) :: pvs) →
+    R !! p = Some w →
+    w = Some v.
+  Proof.
+    intros Hfr Hlookup. specialize (Hfr p v (elem_of_dom_2 _ _ _ Hlookup)).
+    rewrite /first_resolve lookup_insert in Hfr. rewrite Hfr in Hlookup; last done.
+    inversion Hlookup. done.
+  Qed.
+
+End first_resolve.
+
 Section to_proph_map.
   Context (P V : Type) `{Countable P}.
   Implicit Types p : P.