Rename solve_elem_of into set_solver.

It is doing much more than just dealing with ∈, it solves all kinds
of goals involving set operations (including ≡ and ⊆).

theories/collections.v

@@ -207,7 +207,7 @@ Ltac decompose_empty := repeat
 occurrences of [(∪)], [(∩)], [(∖)], [(<$>)], [∅], [{[_]}], [(≡)], and [(⊆)],
 by rewriting these into logically equivalent propositions. For example we
 rewrite [A → x ∈ X ∪ ∅] into [A → x ∈ X ∨ False]. *)
-Ltac unfold_elem_of :=
+Ltac set_unfold :=
   repeat_on_hyps (fun H =>
     repeat match type of H with
     | context [ _ ⊆ _ ] => setoid_rewrite elem_of_subseteq in H
@@ -251,21 +251,21 @@ Ltac unfold_elem_of :=
   end.
 
 (** Since [firstorder] fails or loops on very small goals generated by
-[solve_elem_of] already. We use the [naive_solver] tactic as a substitute.
+[set_solver] already. We use the [naive_solver] tactic as a substitute.
 This tactic either fails or proves the goal. *)
-Tactic Notation "solve_elem_of" tactic3(tac) :=
+Tactic Notation "set_solver" tactic3(tac) :=
   setoid_subst;
   decompose_empty;
-  unfold_elem_of;
+  set_unfold;
   naive_solver tac.
-Tactic Notation "solve_elem_of" "-" hyp_list(Hs) "/" tactic3(tac) :=
-  clear Hs; solve_elem_of tac.
-Tactic Notation "solve_elem_of" "+" hyp_list(Hs) "/" tactic3(tac) :=
-  revert Hs; clear; solve_elem_of tac.
-Tactic Notation "solve_elem_of" := solve_elem_of eauto.
-Tactic Notation "solve_elem_of" "-" hyp_list(Hs) := clear Hs; solve_elem_of.
-Tactic Notation "solve_elem_of" "+" hyp_list(Hs) :=
-  revert Hs; clear; solve_elem_of.
+Tactic Notation "set_solver" "-" hyp_list(Hs) "/" tactic3(tac) :=
+  clear Hs; set_solver tac.
+Tactic Notation "set_solver" "+" hyp_list(Hs) "/" tactic3(tac) :=
+  revert Hs; clear; set_solver tac.
+Tactic Notation "set_solver" := set_solver eauto.
+Tactic Notation "set_solver" "-" hyp_list(Hs) := clear Hs; set_solver.
+Tactic Notation "set_solver" "+" hyp_list(Hs) :=
+  revert Hs; clear; set_solver.
 
 (** * More theorems *)
 Section collection.
@@ -273,7 +273,7 @@ Section collection.
   Implicit Types X Y : C.
 
   Global Instance: Lattice C.
-  Proof. split. apply _. firstorder auto. solve_elem_of. Qed.
+  Proof. split. apply _. firstorder auto. set_solver. Qed.
   Global Instance difference_proper :
      Proper ((≡) ==> (≡) ==> (≡)) (@difference C _).
   Proof.
@@ -281,23 +281,23 @@ Section collection.
     by rewrite !elem_of_difference, HX, HY.
   Qed.
   Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma difference_diag X : X ∖ X ≡ ∅.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y).
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
 
   Section leibniz.
     Context `{!LeibnizEquiv C}.
@@ -334,10 +334,10 @@ Section collection.
     Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅.
     Proof.
       intros [HXY1 HXY2] Hdiff. destruct HXY2. intros x.
-      destruct (decide (x ∈ X)); solve_elem_of.
+      destruct (decide (x ∈ X)); set_solver.
     Qed.
     Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y.
-    Proof. intros ? x ?; apply dec_stable; solve_elem_of. Qed.
+    Proof. intros ? x ?; apply dec_stable; set_solver. Qed.
     Context `{!LeibnizEquiv C}.
     Lemma union_difference_L X Y : X ⊆ Y → Y = X ∪ Y ∖ X.
     Proof. unfold_leibniz. apply union_difference. Qed.
@@ -396,33 +396,33 @@ Section NoDup.
   Proof. firstorder. Qed.
 
   Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X.
-  Proof. unfold elem_of_upto. solve_elem_of. Qed.
+  Proof. unfold elem_of_upto. set_solver. Qed.
   Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅.
-  Proof. unfold elem_of_upto. solve_elem_of. Qed.
+  Proof. unfold elem_of_upto. set_solver. Qed.
   Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y.
-  Proof. unfold elem_of_upto. solve_elem_of. Qed.
+  Proof. unfold elem_of_upto. set_solver. Qed.
 
   Lemma elem_of_upto_union X Y x :
     elem_of_upto x (X ∪ Y) ↔ elem_of_upto x X ∨ elem_of_upto x Y.
-  Proof. unfold elem_of_upto. solve_elem_of. Qed.
+  Proof. unfold elem_of_upto. set_solver. Qed.
   Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y.
-  Proof. unfold elem_of_upto. solve_elem_of. Qed.
+  Proof. unfold elem_of_upto. set_solver. Qed.
 
   Lemma set_NoDup_empty: set_NoDup ∅.
-  Proof. unfold set_NoDup. solve_elem_of. Qed.
+  Proof. unfold set_NoDup. set_solver. Qed.
   Lemma set_NoDup_add x X :
     ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X).
-  Proof. unfold set_NoDup, elem_of_upto. solve_elem_of. Qed.
+  Proof. unfold set_NoDup, elem_of_upto. set_solver. Qed.
   Lemma set_NoDup_inv_add x X :
     x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X.
   Proof.
     intros Hin Hnodup [y [??]].
-    rewrite (Hnodup x y) in Hin; solve_elem_of.
+    rewrite (Hnodup x y) in Hin; set_solver.
   Qed.
   Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X.
-  Proof. unfold set_NoDup. solve_elem_of. Qed.
+  Proof. unfold set_NoDup. set_solver. Qed.
   Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y.
-  Proof. unfold set_NoDup. solve_elem_of. Qed.
+  Proof. unfold set_NoDup. set_solver. Qed.
 End NoDup.
 
 (** * Quantifiers *)
@@ -433,27 +433,27 @@ Section quantifiers.
   Definition set_Exists X := ∃ x, x ∈ X ∧ P x.
 
   Lemma set_Forall_empty : set_Forall ∅.
-  Proof. unfold set_Forall. solve_elem_of. Qed.
+  Proof. unfold set_Forall. set_solver. Qed.
   Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x.
-  Proof. unfold set_Forall. solve_elem_of. Qed.
+  Proof. unfold set_Forall. set_solver. Qed.
   Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y).
-  Proof. unfold set_Forall. solve_elem_of. Qed.
+  Proof. unfold set_Forall. set_solver. Qed.
   Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X.
-  Proof. unfold set_Forall. solve_elem_of. Qed.
+  Proof. unfold set_Forall. set_solver. Qed.
   Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y.
-  Proof. unfold set_Forall. solve_elem_of. Qed.
+  Proof. unfold set_Forall. set_solver. Qed.
 
   Lemma set_Exists_empty : ¬set_Exists ∅.
-  Proof. unfold set_Exists. solve_elem_of. Qed.
+  Proof. unfold set_Exists. set_solver. Qed.
   Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x.
-  Proof. unfold set_Exists. solve_elem_of. Qed.
+  Proof. unfold set_Exists. set_solver. Qed.
   Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y).
-  Proof. unfold set_Exists. solve_elem_of. Qed.
+  Proof. unfold set_Exists. set_solver. Qed.
   Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y).
-  Proof. unfold set_Exists. solve_elem_of. Qed.
+  Proof. unfold set_Exists. set_solver. Qed.
   Lemma set_Exists_union_inv X Y :
     set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y.
-  Proof. unfold set_Exists. solve_elem_of. Qed.
+  Proof. unfold set_Exists. set_solver. Qed.
 End quantifiers.
 
 Section more_quantifiers.
@@ -510,7 +510,7 @@ Section fresh.
   Qed.
   Lemma Forall_fresh_subseteq X Y xs :
     Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs.
-  Proof. rewrite !Forall_fresh_alt; solve_elem_of. Qed.
+  Proof. rewrite !Forall_fresh_alt; set_solver. Qed.
 
   Lemma fresh_list_length n X : length (fresh_list n X) = n.
   Proof. revert X. induction n; simpl; auto. Qed.
@@ -518,12 +518,12 @@ Section fresh.
   Proof.
     revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|].
     rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|].
-    apply IH in Hin; solve_elem_of.
+    apply IH in Hin; set_solver.
   Qed.
   Lemma NoDup_fresh_list n X : NoDup (fresh_list n X).
   Proof.
     revert X. induction n; simpl; constructor; auto.
-    intros Hin; apply fresh_list_is_fresh in Hin; solve_elem_of.
+    intros Hin; apply fresh_list_is_fresh in Hin; set_solver.
   Qed.
   Lemma Forall_fresh_list X n : Forall_fresh X (fresh_list n X).
   Proof.
@@ -537,50 +537,50 @@ Section collection_monad.
 
   Global Instance collection_fmap_mono {A B} :
     Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B).
-  Proof. intros f g ? X Y ?; solve_elem_of. Qed.
+  Proof. intros f g ? X Y ?; set_solver. Qed.
   Global Instance collection_fmap_proper {A B} :
     Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B).
-  Proof. intros f g ? X Y [??]; split; solve_elem_of. Qed.
+  Proof. intros f g ? X Y [??]; split; set_solver. Qed.
   Global Instance collection_bind_mono {A B} :
     Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B).
-  Proof. unfold respectful; intros f g Hfg X Y ?; solve_elem_of. Qed.
+  Proof. unfold respectful; intros f g Hfg X Y ?; set_solver. Qed.
   Global Instance collection_bind_proper {A B} :
     Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B).
-  Proof. unfold respectful; intros f g Hfg X Y [??]; split; solve_elem_of. Qed.
+  Proof. unfold respectful; intros f g Hfg X Y [??]; split; set_solver. Qed.
   Global Instance collection_join_mono {A} :
     Proper ((⊆) ==> (⊆)) (@mjoin M _ A).
-  Proof. intros X Y ?; solve_elem_of. Qed.
+  Proof. intros X Y ?; set_solver. Qed.
   Global Instance collection_join_proper {A} :
     Proper ((≡) ==> (≡)) (@mjoin M _ A).
-  Proof. intros X Y [??]; split; solve_elem_of. Qed.
+  Proof. intros X Y [??]; split; set_solver. Qed.
 
   Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) :
     g ∘ f <$> X ≡ g <$> (f <$> X).
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma elem_of_fmap_1 {A B} (f : A → B) (X : M A) (y : B) :
     y ∈ f <$> X → ∃ x, y = f x ∧ x ∈ X.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma elem_of_fmap_2 {A B} (f : A → B) (X : M A) (x : A) :
     x ∈ X → f x ∈ f <$> X.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
   Lemma elem_of_fmap_2_alt {A B} (f : A → B) (X : M A) (x : A) (y : B) :
     x ∈ X → y = f x → y ∈ f <$> X.
-  Proof. solve_elem_of. Qed.
+  Proof. set_solver. Qed.
 
   Lemma elem_of_mapM {A B} (f : A → M B) l k :
     l ∈ mapM f k ↔ Forall2 (λ x y, x ∈ f y) l k.
   Proof.
     split.
-    - revert l. induction k; solve_elem_of.
-    - induction 1; solve_elem_of.
+    - revert l. induction k; set_solver.
+    - induction 1; set_solver.
   Qed.
   Lemma collection_mapM_length {A B} (f : A → M B) l k :
     l ∈ mapM f k → length l = length k.
-  Proof. revert l; induction k; solve_elem_of. Qed.
+  Proof. revert l; induction k; set_solver. Qed.
   Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k :
     Forall (λ x, ∀ y, y ∈ g x → f y = x) l → k ∈ mapM g l → fmap f k = l.
   Proof.
@@ -606,7 +606,7 @@ Section finite.
   Context `{SimpleCollection A B}.
   Global Instance set_finite_subseteq :
      Proper (flip (⊆) ==> impl) (@set_finite A B _).
-  Proof. intros X Y HX [l Hl]; exists l; solve_elem_of. Qed.
+  Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed.
   Global Instance set_finite_proper : Proper ((≡) ==> iff) (@set_finite A B _).
   Proof. by intros X Y [??]; split; apply set_finite_subseteq. Qed.
   Lemma empty_finite : set_finite ∅.
@@ -619,9 +619,9 @@ Section finite.
     rewrite elem_of_union, elem_of_app; naive_solver.
   Qed.
   Lemma union_finite_inv_l X Y : set_finite (X ∪ Y) → set_finite X.
-  Proof. intros [l ?]; exists l; solve_elem_of. Qed.
+  Proof. intros [l ?]; exists l; set_solver. Qed.
   Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y.
-  Proof. intros [l ?]; exists l; solve_elem_of. Qed.
+  Proof. intros [l ?]; exists l; set_solver. Qed.
 End finite.
 
 Section more_finite.
@@ -636,6 +636,6 @@ Section more_finite.
     set_finite Y → set_finite (X ∖ Y) → set_finite X.
   Proof.
     intros [l ?] [k ?]; exists (l ++ k).
-    intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; solve_elem_of.
+    intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; set_solver.
   Qed.
 End more_finite.

theories/fin_collections.v

@@ -41,7 +41,7 @@ Qed.
 Lemma elements_singleton x : elements {[ x ]} = [x].
 Proof.
   apply Permutation_singleton. by rewrite <-(right_id ∅ (∪) {[x]}),
-    elements_union_singleton, elements_empty by solve_elem_of.
+    elements_union_singleton, elements_empty by set_solver.
 Qed.
 Lemma elements_contains X Y : X ⊆ Y → elements X `contains` elements Y.
 Proof.
@@ -90,7 +90,7 @@ Proof.
   intros E. destruct (size_pos_elem_of X); auto with lia.
   exists x. apply elem_of_equiv. split.
   - rewrite elem_of_singleton. eauto using size_singleton_inv.
-  - solve_elem_of.
+  - set_solver.
 Qed.
 Lemma size_union X Y : X ∩ Y ≡ ∅ → size (X ∪ Y) = size X + size Y.
 Proof.
@@ -98,7 +98,7 @@ Proof.
   apply Permutation_length, NoDup_Permutation.
   - apply NoDup_elements.
   - apply NoDup_app; repeat split; try apply NoDup_elements.
-    intros x; rewrite !elem_of_elements; solve_elem_of.
+    intros x; rewrite !elem_of_elements; set_solver.
   - intros. by rewrite elem_of_app, !elem_of_elements, elem_of_union.
 Qed.
 Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100.
@@ -121,15 +121,15 @@ Next Obligation.
 Qed.
 Lemma size_union_alt X Y : size (X ∪ Y) = size X + size (Y ∖ X).
 Proof.
-  rewrite <-size_union by solve_elem_of.
-  setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by solve_elem_of.
-  rewrite <-union_difference, (comm (∪)); solve_elem_of.
+  rewrite <-size_union by set_solver.
+  setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by set_solver.
+  rewrite <-union_difference, (comm (∪)); set_solver.
 Qed.
 Lemma subseteq_size X Y : X ⊆ Y → size X ≤ size Y.
 Proof. intros. rewrite (union_difference X Y), size_union_alt by done. lia. Qed.
 Lemma subset_size X Y : X ⊂ Y → size X < size Y.
 Proof.
-  intros. rewrite (union_difference X Y) by solve_elem_of.
+  intros. rewrite (union_difference X Y) by set_solver.
   rewrite size_union_alt, difference_twice.
   cut (size (Y ∖ X) ≠ 0); [lia |].
   by apply size_non_empty_iff, non_empty_difference.
@@ -143,8 +143,8 @@ Proof.
   intros ? Hemp Hadd. apply well_founded_induction with (⊂).
   { apply collection_wf. }
   intros X IH. destruct (collection_choose_or_empty X) as [[x ?]|HX].
-  - rewrite (union_difference {[ x ]} X) by solve_elem_of.
-    apply Hadd. solve_elem_of. apply IH; solve_elem_of.
+  - rewrite (union_difference {[ x ]} X) by set_solver.
+    apply Hadd. set_solver. apply IH; set_solver.
   - by rewrite HX.
 Qed.
 Lemma collection_fold_ind {B} (P : B → C → Prop) (f : A → B → B) (b : B) :
@@ -158,10 +158,10 @@ Proof.
     symmetry. apply elem_of_elements. }
   induction 1 as [|x l ?? IH]; simpl.
   - intros X HX. setoid_rewrite elem_of_nil in HX.
-    rewrite equiv_empty. done. solve_elem_of.
+    rewrite equiv_empty. done. set_solver.
   - intros X HX. setoid_rewrite elem_of_cons in HX.
-    rewrite (union_difference {[ x ]} X) by solve_elem_of.
-    apply Hadd. solve_elem_of. apply IH. solve_elem_of.
+    rewrite (union_difference {[ x ]} X) by set_solver.
+    apply Hadd. set_solver. apply IH. set_solver.
 Qed.
 Lemma collection_fold_proper {B} (R : relation B) `{!Equivalence R}
     (f : A → B → B) (b : B) `{!Proper ((=) ==> R ==> R) f}

theories/fin_map_dom.v

@@ -36,13 +36,13 @@ Proof.
 Qed.
 Lemma dom_empty {A} : dom D (@empty (M A) _) ≡ ∅.
 Proof.
-  split; intro; [|solve_elem_of].
+  split; intro; [|set_solver].
   rewrite elem_of_dom, lookup_empty. by inversion 1.
 Qed.
 Lemma dom_empty_inv {A} (m : M A) : dom D m ≡ ∅ → m = ∅.
 Proof.
   intros E. apply map_empty. intros. apply not_elem_of_dom.
-  rewrite E. solve_elem_of.
+  rewrite E. set_solver.
 Qed.
 Lemma dom_alter {A} f (m : M A) i : dom D (alter f i m) ≡ dom D m.
 Proof.
@@ -54,19 +54,19 @@ Lemma dom_insert {A} (m : M A) i x : dom D (<[i:=x]>m) ≡ {[ i ]} ∪ dom D m.
 Proof.
   apply elem_of_equiv. intros j. rewrite elem_of_union, !elem_of_dom.
   unfold is_Some. setoid_rewrite lookup_insert_Some.
-  destruct (decide (i = j)); solve_elem_of.
+  destruct (decide (i = j)); set_solver.
 Qed.
 Lemma dom_insert_subseteq {A} (m : M A) i x : dom D m ⊆ dom D (<[i:=x]>m).
-Proof. rewrite (dom_insert _). solve_elem_of. Qed.
+Proof. rewrite (dom_insert _). set_solver. Qed.
 Lemma dom_insert_subseteq_compat_l {A} (m : M A) i x X :
   X ⊆ dom D m → X ⊆ dom D (<[i:=x]>m).
 Proof. intros. transitivity (dom D m); eauto using dom_insert_subseteq. Qed.
 Lemma dom_singleton {A} (i : K) (x : A) : dom D {[i := x]} ≡ {[ i ]}.
-Proof. rewrite <-insert_empty, dom_insert, dom_empty; solve_elem_of. Qed.
+Proof. rewrite <-insert_empty, dom_insert, dom_empty; set_solver. Qed.
 Lemma dom_delete {A} (m : M A) i : dom D (delete i m) ≡ dom D m ∖ {[ i ]}.
 Proof.
   apply elem_of_equiv. intros j. rewrite elem_of_difference, !elem_of_dom.
-  unfold is_Some. setoid_rewrite lookup_delete_Some. solve_elem_of.
+  unfold is_Some. setoid_rewrite lookup_delete_Some. set_solver.
 Qed.
 Lemma delete_partial_alter_dom {A} (m : M A) i f :
   i ∉ dom D m → delete i (partial_alter f i m) = m.

theories/hashset.v

@@ -155,7 +155,7 @@ Proof.
   - revert x. induction l as [|y l IH]; intros x; simpl.
     { by rewrite elem_of_empty. }
     rewrite elem_of_union, elem_of_singleton. intros [->|]; [left|right]; eauto.
-  - induction 1; solve_elem_of.
+  - induction 1; set_solver.
 Qed.
 Lemma NoDup_remove_dups_fast l : NoDup (remove_dups_fast l).
 Proof.