Tweak some proof using tweaks for setoid stuff.

theories/prelude/fin_maps.v

@@ -119,13 +119,13 @@ Context `{FinMap K M}.
 (** ** Setoids *)
 Section setoid.
   Context `{Equiv A}.
-  
+
   Lemma map_equiv_lookup_l (m1 m2 : M A) i x :
     m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y.
   Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed.
 
-  Context `{!Equivalence ((≡) : relation A)}.
-  Global Instance map_equivalence : Equivalence ((≡) : relation (M A)).
+  Global Instance map_equivalence :
+    Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (M A)).
   Proof.
     split.
     - by intros m i.
@@ -147,7 +147,10 @@ Section setoid.
   Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
   Global Instance singleton_proper k :
     Proper ((≡) ==> (≡)) (singletonM k : A → M A).
-  Proof. by intros ???; apply insert_proper. Qed.
+  Proof.
+    intros ???; apply insert_proper; [done|].
+    intros ?. rewrite lookup_empty; constructor.
+  Qed.
   Global Instance delete_proper (i : K) :
     Proper ((≡) ==> (≡)) (delete (M:=M A) i).
   Proof. by apply partial_alter_proper; [constructor|]. Qed.
@@ -170,14 +173,12 @@ Section setoid.
     by do 2 destruct 1; first [apply Hf | constructor].
   Qed.
   Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
-  Proof.
-    intros m1 m2 Hm; apply map_eq; intros i.
-    by unfold_leibniz; apply lookup_proper.
-  Qed.
+  Proof. intros m1 m2 Hm; apply map_eq; intros i. apply leibniz_equiv, Hm. Qed.
   Lemma map_equiv_empty (m : M A) : m ≡ ∅ ↔ m = ∅.
   Proof.
-    split; [intros Hm; apply map_eq; intros i|by intros ->].
-    by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty.
+    split; [intros Hm; apply map_eq; intros i|intros ->].
+    - generalize (Hm i). by rewrite lookup_empty, equiv_None.
+    - intros ?. rewrite lookup_empty; constructor.
   Qed.
   Global Instance map_fmap_proper `{Equiv B} (f : A → B) :
     Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (fmap (M:=M) f).

theories/prelude/list.v

@@ -2753,9 +2753,8 @@ Section setoid.
     by setoid_rewrite equiv_option_Forall2.
   Qed.
 
-  Context {Hequiv: Equivalence ((≡) : relation A)}.
-
-  Global Instance list_equivalence : Equivalence ((≡) : relation (list A)).
+  Global Instance list_equivalence :
+    Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (list A)).
   Proof.
     split.
     - intros l. by apply equiv_Forall2.
@@ -2766,48 +2765,53 @@ Section setoid.
   Proof. induction 1; f_equal; fold_leibniz; auto. Qed.
 
   Global Instance cons_proper : Proper ((≡) ==> (≡) ==> (≡)) (@cons A).
-  Proof using -(Hequiv). by constructor. Qed.
+  Proof. by constructor. Qed.
   Global Instance app_proper : Proper ((≡) ==> (≡) ==> (≡)) (@app A).
-  Proof using -(Hequiv). induction 1; intros ???; simpl; try constructor; auto. Qed.
+  Proof. induction 1; intros ???; simpl; try constructor; auto. Qed.
   Global Instance length_proper : Proper ((≡) ==> (=)) (@length A).
-  Proof using -(Hequiv). induction 1; f_equal/=; auto. Qed.
+  Proof. induction 1; f_equal/=; auto. Qed.
   Global Instance tail_proper : Proper ((≡) ==> (≡)) (@tail A).
-  Proof. by destruct 1. Qed.
+  Proof. destruct 1; try constructor; auto. Qed.
   Global Instance take_proper n : Proper ((≡) ==> (≡)) (@take A n).
-  Proof using -(Hequiv). induction n; destruct 1; constructor; auto. Qed.
+  Proof. induction n; destruct 1; constructor; auto. Qed.
   Global Instance drop_proper n : Proper ((≡) ==> (≡)) (@drop A n).
-  Proof using -(Hequiv). induction n; destruct 1; simpl; try constructor; auto. Qed.
+  Proof. induction n; destruct 1; simpl; try constructor; auto. Qed.
   Global Instance list_lookup_proper i :
     Proper ((≡) ==> (≡)) (lookup (M:=list A) i).
-  Proof. induction i; destruct 1; simpl; f_equiv; auto. Qed.
+  Proof. induction i; destruct 1; simpl; try constructor; auto. Qed.
   Global Instance list_alter_proper f i :
     Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (alter (M:=list A) f i).
-  Proof using -(Hequiv). intros. induction i; destruct 1; constructor; eauto. Qed.
+  Proof. intros. induction i; destruct 1; constructor; eauto. Qed.
   Global Instance list_insert_proper i :
     Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=list A) i).
-  Proof using -(Hequiv). intros ???; induction i; destruct 1; constructor; eauto. Qed.
+  Proof. intros ???; induction i; destruct 1; constructor; eauto. Qed.
   Global Instance list_inserts_proper i :
     Proper ((≡) ==> (≡) ==> (≡)) (@list_inserts A i).
-  Proof using -(Hequiv).
+  Proof.
     intros k1 k2 Hk; revert i.
     induction Hk; intros ????; simpl; try f_equiv; naive_solver.
   Qed.
   Global Instance list_delete_proper i :
     Proper ((≡) ==> (≡)) (delete (M:=list A) i).
-  Proof using -(Hequiv). induction i; destruct 1; try constructor; eauto. Qed.
+  Proof. induction i; destruct 1; try constructor; eauto. Qed.
   Global Instance option_list_proper : Proper ((≡) ==> (≡)) (@option_list A).
-  Proof. destruct 1; by constructor. Qed.
+  Proof. destruct 1; repeat constructor; auto. Qed.
   Global Instance list_filter_proper P `{∀ x, Decision (P x)} :
     Proper ((≡) ==> iff) P → Proper ((≡) ==> (≡)) (filter (B:=list A) P).
-  Proof using -(Hequiv). intros ???. rewrite !equiv_Forall2. by apply Forall2_filter. Qed.
+  Proof. intros ???. rewrite !equiv_Forall2. by apply Forall2_filter. Qed.
   Global Instance replicate_proper n : Proper ((≡) ==> (≡)) (@replicate A n).
-  Proof using -(Hequiv). induction n; constructor; auto. Qed.
+  Proof. induction n; constructor; auto. Qed.
   Global Instance reverse_proper : Proper ((≡) ==> (≡)) (@reverse A).
-  Proof. induction 1; rewrite ?reverse_cons; repeat (done || f_equiv). Qed.
+  Proof.
+    induction 1; rewrite ?reverse_cons; simpl; [constructor|].
+    apply app_proper; repeat constructor; auto.
+  Qed.
   Global Instance last_proper : Proper ((≡) ==> (≡)) (@last A).
-  Proof. induction 1 as [|????? []]; simpl; repeat (done || f_equiv). Qed.
+  Proof. induction 1 as [|????? []]; simpl; repeat constructor; auto. Qed.
   Global Instance resize_proper n : Proper ((≡) ==> (≡) ==> (≡)) (@resize A n).
-  Proof. induction n; destruct 2; simpl; repeat (auto || f_equiv). Qed.
+  Proof.
+    induction n; destruct 2; simpl; repeat (constructor || f_equiv); auto.
+  Qed.
 End setoid.
 
 (** * Properties of the monadic operations *)

theories/prelude/option.v

@@ -115,36 +115,38 @@ End Forall2.
 Instance option_equiv `{Equiv A} : Equiv (option A) := option_Forall2 (≡).
 
 Section setoids.
-  Context `{Equiv A} {Hequiv: Equivalence ((≡) : relation A)}.
+  Context `{Equiv A}.
   Implicit Types mx my : option A.
 
   Lemma equiv_option_Forall2 mx my : mx ≡ my ↔ option_Forall2 (≡) mx my.
-  Proof using -(Hequiv). done. Qed.
+  Proof. done. Qed.
 
-  Global Instance option_equivalence : Equivalence ((≡) : relation (option A)).
+  Global Instance option_equivalence :
+    Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (option A)).
   Proof. apply _. Qed.
   Global Instance Some_proper : Proper ((≡) ==> (≡)) (@Some A).
-  Proof using -(Hequiv). by constructor. Qed.
+  Proof. by constructor. Qed.
   Global Instance Some_equiv_inj : Inj (≡) (≡) (@Some A).
-  Proof using -(Hequiv). by inversion_clear 1. Qed.
+  Proof. by inversion_clear 1. Qed.
   Global Instance option_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (option A).
-  Proof. intros x y; destruct 1; fold_leibniz; congruence. Qed.
+  Proof. intros x y; destruct 1; f_equal; by apply leibniz_equiv. Qed.
 
   Lemma equiv_None mx : mx ≡ None ↔ mx = None.
-  Proof. split; [by inversion_clear 1|by intros ->]. Qed.
+  Proof. split; [by inversion_clear 1|intros ->; constructor]. Qed.
   Lemma equiv_Some_inv_l mx my x :
     mx ≡ my → mx = Some x → ∃ y, my = Some y ∧ x ≡ y.
-  Proof using -(Hequiv). destruct 1; naive_solver. Qed.
+  Proof. destruct 1; naive_solver. Qed.
   Lemma equiv_Some_inv_r mx my y :
     mx ≡ my → my = Some y → ∃ x, mx = Some x ∧ x ≡ y.
-  Proof using -(Hequiv). destruct 1; naive_solver. Qed.
+  Proof. destruct 1; naive_solver. Qed.
   Lemma equiv_Some_inv_l' my x : Some x ≡ my → ∃ x', Some x' = my ∧ x ≡ x'.
-  Proof using -(Hequiv). intros ?%(equiv_Some_inv_l _ _ x); naive_solver. Qed.
-  Lemma equiv_Some_inv_r' mx y : mx ≡ Some y → ∃ y', mx = Some y' ∧ y ≡ y'.
+  Proof. intros ?%(equiv_Some_inv_l _ _ x); naive_solver. Qed.
+  Lemma equiv_Some_inv_r' `{!Equivalence ((≡) : relation A)} mx y :
+    mx ≡ Some y → ∃ y', mx = Some y' ∧ y ≡ y'.
   Proof. intros ?%(equiv_Some_inv_r _ _ y); naive_solver. Qed.
 
   Global Instance is_Some_proper : Proper ((≡) ==> iff) (@is_Some A).
-  Proof using -(Hequiv). inversion_clear 1; split; eauto. Qed.
+  Proof. inversion_clear 1; split; eauto. Qed.
   Global Instance from_option_proper {B} (R : relation B) (f : A → B) :
     Proper ((≡) ==> R) f → Proper (R ==> (≡) ==> R) (from_option f).
   Proof. destruct 3; simpl; auto. Qed.