tune "Proof using" directives to minimize differences to previous types of all lemmas

theories/countable.v

@@ -32,7 +32,7 @@ Qed.
 
 (** * Choice principles *)
 Section choice.
-  Context `{Countable A} (P : A → Prop) `{∀ x, Decision (P x)}.
+  Context `{Countable A} (P : A → Prop).
 
   Inductive choose_step: relation positive :=
     | choose_step_None {p} : decode p = None → choose_step (Psucc p) p
@@ -50,6 +50,9 @@ Section choice.
     constructor. intros j.
     inversion 1 as [? Hd|? y Hd]; subst; auto with lia.
   Qed.
+
+  Context `{∀ x, Decision (P x)}.
+
   Fixpoint choose_go {i} (acc : Acc choose_step i) : A :=
     match Some_dec (decode i) with
     | inleft (x↾Hx) =>

theories/fin_maps.v

@@ -118,7 +118,13 @@ Context `{FinMap K M}.
 
 (** ** Setoids *)
 Section setoid.
-  Context `{Equiv A} `{!Equivalence ((≡) : relation A)}.
+  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)).
   Proof.
     split.
@@ -173,9 +179,6 @@ Section setoid.
     split; [intros Hm; apply map_eq; intros i|by intros ->].
     by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty.
   Qed.
-  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.
   Global Instance map_fmap_proper `{Equiv B} (f : A → B) :
     Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (fmap (M:=M) f).
   Proof.

theories/finite.v

@@ -171,13 +171,15 @@ Proof. apply finite_bijective. eauto. Qed.
 
 (** Decidability of quantification over finite types *)
 Section forall_exists.
-  Context `{Finite A} (P : A → Prop) `{∀ x, Decision (P x)}.
+  Context `{Finite A} (P : A → Prop).
 
   Lemma Forall_finite : Forall P (enum A) ↔ (∀ x, P x).
   Proof. rewrite Forall_forall. intuition auto using elem_of_enum. Qed.
   Lemma Exists_finite : Exists P (enum A) ↔ (∃ x, P x).
   Proof. rewrite Exists_exists. naive_solver eauto using elem_of_enum. Qed.
 
+  Context `{∀ x, Decision (P x)}.
+
   Global Instance forall_dec: Decision (∀ x, P x).
   Proof.
    refine (cast_if (decide (Forall P (enum A))));

theories/list.v

@@ -735,6 +735,28 @@ End no_dup_dec.
 
 (** ** Set operations on lists *)
 Section list_set.
+  Lemma elem_of_list_intersection_with f l k x :
+    x ∈ list_intersection_with f l k ↔ ∃ x1 x2,
+        x1 ∈ l ∧ x2 ∈ k ∧ f x1 x2 = Some x.
+  Proof.
+    split.
+    - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
+      intros Hx. setoid_rewrite elem_of_cons.
+      cut ((∃ x2, x2 ∈ k ∧ f x1 x2 = Some x)
+           ∨ x ∈ list_intersection_with f l k); [naive_solver|].
+      clear IH. revert Hx. generalize (list_intersection_with f l k).
+      induction k; simpl; [by auto|].
+      case_match; setoid_rewrite elem_of_cons; naive_solver.
+    - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
+      + generalize (list_intersection_with f l k).
+        induction Hx2; simpl; [by rewrite Hx; left |].
+        case_match; simpl; try setoid_rewrite elem_of_cons; auto.
+      + generalize (IH Hx). clear Hx IH Hx2.
+        generalize (list_intersection_with f l k).
+        induction k; simpl; intros; [done|].
+        case_match; simpl; rewrite ?elem_of_cons; auto.
+  Qed.
+
   Context `{!EqDecision A}.
   Lemma elem_of_list_difference l k x : x ∈ list_difference l k ↔ x ∈ l ∧ x ∉ k.
   Proof.
@@ -773,27 +795,6 @@ Section list_set.
     - constructor. rewrite elem_of_list_intersection; intuition. done.
     - done.
   Qed.
-  Lemma elem_of_list_intersection_with f l k x :
-    x ∈ list_intersection_with f l k ↔ ∃ x1 x2,
-      x1 ∈ l ∧ x2 ∈ k ∧ f x1 x2 = Some x.
-  Proof.
-    split.
-    - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
-      intros Hx. setoid_rewrite elem_of_cons.
-      cut ((∃ x2, x2 ∈ k ∧ f x1 x2 = Some x)
-        ∨ x ∈ list_intersection_with f l k); [naive_solver|].
-      clear IH. revert Hx. generalize (list_intersection_with f l k).
-      induction k; simpl; [by auto|].
-      case_match; setoid_rewrite elem_of_cons; naive_solver.
-    - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
-      + generalize (list_intersection_with f l k).
-        induction Hx2; simpl; [by rewrite Hx; left |].
-        case_match; simpl; try setoid_rewrite elem_of_cons; auto.
-      + generalize (IH Hx). clear Hx IH Hx2.
-        generalize (list_intersection_with f l k).
-        induction k; simpl; intros; [done|].
-        case_match; simpl; rewrite ?elem_of_cons; auto.
-  Qed.
 End list_set.
 
 (** ** Properties of the [filter] function *)
@@ -2171,7 +2172,7 @@ Section Forall_Exists.
   Lemma Forall_replicate n x : P x → Forall P (replicate n x).
   Proof. induction n; simpl; constructor; auto. Qed.
   Lemma Forall_replicate_eq n (x : A) : Forall (x =) (replicate n x).
-  Proof. induction n; simpl; constructor; auto. Qed.
+  Proof using -(P). induction n; simpl; constructor; auto. Qed.
   Lemma Forall_take n l : Forall P l → Forall P (take n l).
   Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed.
   Lemma Forall_drop n l : Forall P l → Forall P (drop n l).
@@ -2741,7 +2742,7 @@ End Forall3.
 
 (** Setoids *)
 Section setoid.
-  Context `{Equiv A} `{!Equivalence ((≡) : relation A)}.
+  Context `{Equiv A}.
   Implicit Types l k : list A.
 
   Lemma equiv_Forall2 l k : l ≡ k ↔ Forall2 (≡) l k.
@@ -2752,6 +2753,8 @@ Section setoid.
     by setoid_rewrite equiv_option_Forall2.
   Qed.
 
+  Context {Hequiv: Equivalence ((≡) : relation A)}.
+
   Global Instance list_equivalence : Equivalence ((≡) : relation (list A)).
   Proof.
     split.
@@ -2763,42 +2766,42 @@ Section setoid.
   Proof. induction 1; f_equal; fold_leibniz; auto. Qed.
 
   Global Instance cons_proper : Proper ((≡) ==> (≡) ==> (≡)) (@cons A).
-  Proof. by constructor. Qed.
+  Proof using -(Hequiv). by constructor. Qed.
   Global Instance app_proper : Proper ((≡) ==> (≡) ==> (≡)) (@app A).
-  Proof. induction 1; intros ???; simpl; try constructor; auto. Qed.
+  Proof using -(Hequiv). induction 1; intros ???; simpl; try constructor; auto. Qed.
   Global Instance length_proper : Proper ((≡) ==> (=)) (@length A).
-  Proof. induction 1; f_equal/=; auto. Qed.
+  Proof using -(Hequiv). induction 1; f_equal/=; auto. Qed.
   Global Instance tail_proper : Proper ((≡) ==> (≡)) (@tail A).
   Proof. by destruct 1. Qed.
   Global Instance take_proper n : Proper ((≡) ==> (≡)) (@take A n).
-  Proof. induction n; destruct 1; constructor; auto. Qed.
+  Proof using -(Hequiv). induction n; destruct 1; constructor; auto. Qed.
   Global Instance drop_proper n : Proper ((≡) ==> (≡)) (@drop A n).
-  Proof. induction n; destruct 1; simpl; try constructor; auto. Qed.
+  Proof using -(Hequiv). 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.
   Global Instance list_alter_proper f i :
     Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (alter (M:=list A) f i).
-  Proof. intros. induction i; destruct 1; constructor; eauto. Qed.
+  Proof using -(Hequiv). intros. induction i; destruct 1; constructor; eauto. Qed.
   Global Instance list_insert_proper i :
     Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=list A) i).
-  Proof. intros ???; induction i; destruct 1; constructor; eauto. Qed.
+  Proof using -(Hequiv). intros ???; induction i; destruct 1; constructor; eauto. Qed.
   Global Instance list_inserts_proper i :
     Proper ((≡) ==> (≡) ==> (≡)) (@list_inserts A i).
-  Proof.
+  Proof using -(Hequiv).
     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. induction i; destruct 1; try constructor; eauto. Qed.
+  Proof using -(Hequiv). induction i; destruct 1; try constructor; eauto. Qed.
   Global Instance option_list_proper : Proper ((≡) ==> (≡)) (@option_list A).
   Proof. destruct 1; by constructor. Qed.
   Global Instance list_filter_proper P `{∀ x, Decision (P x)} :
     Proper ((≡) ==> iff) P → Proper ((≡) ==> (≡)) (filter (B:=list A) P).
-  Proof. intros ???. rewrite !equiv_Forall2. by apply Forall2_filter. Qed.
+  Proof using -(Hequiv). intros ???. rewrite !equiv_Forall2. by apply Forall2_filter. Qed.
   Global Instance replicate_proper n : Proper ((≡) ==> (≡)) (@replicate A n).
-  Proof. induction n; constructor; auto. Qed.
+  Proof using -(Hequiv). induction n; constructor; auto. Qed.
   Global Instance reverse_proper : Proper ((≡) ==> (≡)) (@reverse A).
   Proof. induction 1; rewrite ?reverse_cons; repeat (done || f_equiv). Qed.
   Global Instance last_proper : Proper ((≡) ==> (≡)) (@last A).

theories/option.v

@@ -115,18 +115,18 @@ End Forall2.
 Instance option_equiv `{Equiv A} : Equiv (option A) := option_Forall2 (≡).
 
 Section setoids.
-  Context `{Equiv A} `{!Equivalence ((≡) : relation A)}.
+  Context `{Equiv A} {Hequiv: Equivalence ((≡) : relation A)}.
   Implicit Types mx my : option A.
 
   Lemma equiv_option_Forall2 mx my : mx ≡ my ↔ option_Forall2 (≡) mx my.
-  Proof. done. Qed.
+  Proof using -(Hequiv). done. Qed.
 
   Global Instance option_equivalence : Equivalence ((≡) : relation (option A)).
   Proof. apply _. Qed.
   Global Instance Some_proper : Proper ((≡) ==> (≡)) (@Some A).
-  Proof. by constructor. Qed.
+  Proof using -(Hequiv). by constructor. Qed.
   Global Instance Some_equiv_inj : Inj (≡) (≡) (@Some A).
-  Proof. by inversion_clear 1. Qed.
+  Proof using -(Hequiv). by inversion_clear 1. Qed.
   Global Instance option_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (option A).
   Proof. intros x y; destruct 1; fold_leibniz; congruence. Qed.
 
@@ -134,17 +134,17 @@ Section setoids.
   Proof. split; [by inversion_clear 1|by intros ->]. Qed.
   Lemma equiv_Some_inv_l mx my x :
     mx ≡ my → mx = Some x → ∃ y, my = Some y ∧ x ≡ y.
-  Proof. destruct 1; naive_solver. Qed.
+  Proof using -(Hequiv). 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. destruct 1; naive_solver. Qed.
+  Proof using -(Hequiv). destruct 1; naive_solver. Qed.
   Lemma equiv_Some_inv_l' my x : Some x ≡ my → ∃ x', Some x' = my ∧ x ≡ x'.
-  Proof. intros ?%(equiv_Some_inv_l _ _ x); naive_solver. Qed.
+  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_r _ _ y); naive_solver. Qed.
 
   Global Instance is_Some_proper : Proper ((≡) ==> iff) (@is_Some A).
-  Proof. inversion_clear 1; split; eauto. Qed.
+  Proof using -(Hequiv). 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.