Remove elem_of_tactic that uses intuition.

This one (previously solve_elem_of) was hardly used. The tactic that uses
naive_solver (previously esolve_elem_of, now solve_elem_of) has been
extended with flags to say which hypotheses should be cleared/kept.

theories/collections.v

@@ -248,27 +248,22 @@ Ltac unfold_elem_of :=
   | |- context [ _ ∈ of_list _ ] => setoid_rewrite elem_of_of_list
   end.
 
-(** The tactic [solve_elem_of tac] composes the above tactic with [intuition].
-For goals that do not involve [≡], [⊆], [map], or quantifiers this tactic is
-generally powerful enough. This tactic either fails or proves the goal. *)
+(** Since [firstorder] fails or loops on very small goals generated by
+[solve_elem_of] 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) :=
-  setoid_subst;
-  decompose_empty;
-  unfold_elem_of;
-  solve [intuition (simplify_equality; tac)].
-Tactic Notation "solve_elem_of" := solve_elem_of auto.
-
-(** For goals with quantifiers we could use the above tactic but with
-[firstorder] instead of [intuition] as finishing tactic. However, [firstorder]
-fails or loops on very small goals generated by [solve_elem_of] already. We
-use the [naive_solver] tactic as a substitute. This tactic either fails or
-proves the goal. *)
-Tactic Notation "esolve_elem_of" tactic3(tac) :=
   setoid_subst;
   decompose_empty;
   unfold_elem_of;
   naive_solver tac.
-Tactic Notation "esolve_elem_of" := esolve_elem_of eauto.
+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.
 
 (** * More theorems *)
 Section collection.
@@ -284,21 +279,21 @@ Section collection.
     by rewrite !elem_of_difference, HX, HY.
   Qed.
   Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma difference_diag X : X ∖ X ≡ ∅.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y).
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
 
   Section leibniz.
     Context `{!LeibnizEquiv C}.
@@ -335,10 +330,10 @@ 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)); esolve_elem_of.
+      destruct (decide (x ∈ X)); solve_elem_of.
     Qed.
     Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y.
-    Proof. intros ? x ?; apply dec_stable; esolve_elem_of. Qed.
+    Proof. intros ? x ?; apply dec_stable; solve_elem_of. Qed.
     Context `{!LeibnizEquiv C}.
     Lemma union_difference_L X Y : X ⊆ Y → Y = X ∪ Y ∖ X.
     Proof. unfold_leibniz. apply union_difference. Qed.
@@ -397,23 +392,23 @@ 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. esolve_elem_of. Qed.
+  Proof. unfold elem_of_upto. solve_elem_of. Qed.
   Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅.
-  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
+  Proof. unfold elem_of_upto. solve_elem_of. Qed.
   Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y.
-  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
+  Proof. unfold elem_of_upto. solve_elem_of. 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. esolve_elem_of. Qed.
+  Proof. unfold elem_of_upto. solve_elem_of. Qed.
   Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y.
-  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
+  Proof. unfold elem_of_upto. solve_elem_of. Qed.
 
   Lemma set_NoDup_empty: set_NoDup ∅.
   Proof. unfold set_NoDup. solve_elem_of. 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. esolve_elem_of. Qed.
+  Proof. unfold set_NoDup, elem_of_upto. solve_elem_of. Qed.
   Lemma set_NoDup_inv_add x X :
     x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X.
   Proof.
@@ -445,16 +440,16 @@ Section quantifiers.
   Proof. unfold set_Forall. solve_elem_of. Qed.
 
   Lemma set_Exists_empty : ¬set_Exists ∅.
-  Proof. unfold set_Exists. esolve_elem_of. Qed.
+  Proof. unfold set_Exists. solve_elem_of. Qed.
   Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x.
-  Proof. unfold set_Exists. esolve_elem_of. Qed.
+  Proof. unfold set_Exists. solve_elem_of. Qed.
   Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y).
-  Proof. unfold set_Exists. esolve_elem_of. Qed.
+  Proof. unfold set_Exists. solve_elem_of. Qed.
   Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y).
-  Proof. unfold set_Exists. esolve_elem_of. Qed.
+  Proof. unfold set_Exists. solve_elem_of. Qed.
   Lemma set_Exists_union_inv X Y :
     set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y.
-  Proof. unfold set_Exists. esolve_elem_of. Qed.
+  Proof. unfold set_Exists. solve_elem_of. Qed.
 End quantifiers.
 
 Section more_quantifiers.
@@ -511,7 +506,7 @@ 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; esolve_elem_of. Qed.
+  Proof. rewrite !Forall_fresh_alt; solve_elem_of. Qed.
 
   Lemma fresh_list_length n X : length (fresh_list n X) = n.
   Proof. revert X. induction n; simpl; auto. Qed.
@@ -538,41 +533,41 @@ Section collection_monad.
 
   Global Instance collection_fmap_proper {A B} :
     Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B).
-  Proof. intros f g ? X Y [??]; split; esolve_elem_of. Qed.
+  Proof. intros f g ? X Y [??]; split; solve_elem_of. Qed.
   Global Instance collection_bind_proper {A B} :
     Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B).
-  Proof. unfold respectful; intros f g Hfg X Y [??]; split; esolve_elem_of. Qed.
+  Proof. unfold respectful; intros f g Hfg X Y [??]; split; solve_elem_of. Qed.
   Global Instance collection_join_proper {A} :
     Proper ((≡) ==> (≡)) (@mjoin M _ A).
-  Proof. intros X Y [??]; split; esolve_elem_of. Qed.
+  Proof. intros X Y [??]; split; solve_elem_of. Qed.
 
   Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) :
     g ∘ f <$> X ≡ g <$> (f <$> X).
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma elem_of_fmap_1 {A B} (f : A → B) (X : M A) (y : B) :
     y ∈ f <$> X → ∃ x, y = f x ∧ x ∈ X.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma elem_of_fmap_2 {A B} (f : A → B) (X : M A) (x : A) :
     x ∈ X → f x ∈ f <$> X.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. Qed.
   Lemma elem_of_fmap_2_alt {A B} (f : A → B) (X : M A) (x : A) (y : B) :
     x ∈ X → y = f x → y ∈ f <$> X.
-  Proof. esolve_elem_of. Qed.
+  Proof. solve_elem_of. 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; esolve_elem_of.
-    * induction 1; esolve_elem_of.
+    * revert l. induction k; solve_elem_of.
+    * induction 1; solve_elem_of.
   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; esolve_elem_of. Qed.
+  Proof. revert l; induction k; solve_elem_of. 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.
@@ -606,9 +601,9 @@ 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; esolve_elem_of. Qed.
+  Proof. intros [l ?]; exists l; solve_elem_of. Qed.
   Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y.
-  Proof. intros [l ?]; exists l; esolve_elem_of. Qed.
+  Proof. intros [l ?]; exists l; solve_elem_of. Qed.
 End finite.
 
 Section more_finite.

theories/fin_collections.v

@@ -83,7 +83,7 @@ Proof.
   apply Permutation_length, NoDup_Permutation.
   * apply NoDup_elements.
   * apply NoDup_app; repeat split; try apply NoDup_elements.
-    intros x; rewrite !elem_of_elements; esolve_elem_of.
+    intros x; rewrite !elem_of_elements; solve_elem_of.
   * 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.
@@ -107,7 +107,7 @@ 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 esolve_elem_of.
+  setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by solve_elem_of.
   rewrite <-union_difference, (commutative (∪)); solve_elem_of.
 Qed.
 Lemma subseteq_size X Y : X ⊆ Y → size X ≤ size Y.
@@ -129,7 +129,7 @@ Proof.
   { 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. esolve_elem_of.
+    apply Hadd. solve_elem_of. apply IH; solve_elem_of.
   * by rewrite HX.
 Qed.
 Lemma collection_fold_ind {B} (P : B → C → Prop) (f : A → B → B) (b : B) :
@@ -143,10 +143,10 @@ 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. esolve_elem_of.
+    rewrite equiv_empty. done. solve_elem_of.
   * intros X HX. setoid_rewrite elem_of_cons in HX.
-    rewrite (union_difference {[ x ]} X) by esolve_elem_of.
-    apply Hadd. solve_elem_of. apply IH. esolve_elem_of.
+    rewrite (union_difference {[ x ]} X) by solve_elem_of.
+    apply Hadd. solve_elem_of. apply IH. solve_elem_of.
 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

@@ -54,7 +54,7 @@ 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)); esolve_elem_of.
+  destruct (decide (i = j)); solve_elem_of.
 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.
@@ -66,7 +66,7 @@ Proof. rewrite <-insert_empty, dom_insert, dom_empty; solve_elem_of. 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. esolve_elem_of.
+  unfold is_Some. setoid_rewrite lookup_delete_Some. solve_elem_of.
 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; esolve_elem_of.
+  * induction 1; solve_elem_of.
 Qed.
 Lemma NoDup_remove_dups_fast l : NoDup (remove_dups_fast l).
 Proof.