write a tactic that can solve Proper goals, and use it in a few places

This replaces f_equiv and solve_proper with our own, hopefully better, versions

algebra/cofe.v

@@ -40,8 +40,6 @@ Tactic Notation "cofe_subst" :=
   | H:@dist ?A ?d ?n _ ?x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x
   end.
 
-Tactic Notation "solve_ne" := intros; solve_proper.
-
 Record chain (A : Type) `{Dist A} := {
   chain_car :> nat → A;
   chain_cauchy n i : n < i → chain_car i ≡{n}≡ chain_car (S n)

algebra/sts.v

@@ -77,7 +77,7 @@ Proof. by intros ??? ?? [??]; split; apply up_preserving. Qed.
 Global Instance up_set_preserving : Proper ((⊆) ==> flip (⊆) ==> (⊆)) up_set.
 Proof.
   intros S1 S2 HS T1 T2 HT. rewrite /up_set.
-  f_equiv; last done. move =>s1 s2 Hs. simpl in HT. by apply up_preserving.
+  f_equiv. move =>s1 s2 Hs. simpl in HT. by apply up_preserving.
 Qed.
 Global Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set.
 Proof. by intros S1 S2 [??] T1 T2 [??]; split; apply up_set_preserving. Qed.

algebra/upred.v

@@ -648,7 +648,7 @@ Proof. intros; apply (anti_symm _); auto using const_intro. Qed.
 Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊑ (a ≡ b).
 Proof. intros ->; apply eq_refl. Qed.
 Lemma eq_sym {A : cofeT} (a b : A) : (a ≡ b) ⊑ (b ≡ a).
-Proof. apply (eq_rewrite a b (λ b, b ≡ a)%I); auto using eq_refl. solve_ne. Qed.
+Proof. apply (eq_rewrite a b (λ b, b ≡ a)%I); auto using eq_refl. solve_proper. Qed.
 
 (* BI connectives *)
 Lemma sep_mono P P' Q Q' : P ⊑ Q → P' ⊑ Q' → (P ★ P') ⊑ (Q ★ Q').

barrier/proof.v

@@ -52,19 +52,17 @@ Definition recv (l : loc) (R : iProp) : iProp :=
     saved_prop_own i Q ★ ▷ (Q -★ R))%I.
 
 (** Setoids *)
-(* TODO: These lemmas really ought to be doable by just calling a tactic.
-   It is just application of already registered congruence lemmas. *)
 Global Instance waiting_ne n : Proper (dist n ==> (=) ==> dist n) waiting.
-Proof. intros P P' HP I ? <-. rewrite /waiting. by setoid_rewrite HP. Qed.
+Proof. solve_proper. Qed.
 Global Instance barrier_inv_ne n l :
-  Proper (dist n ==> pointwise_relation _ (dist n)) (barrier_inv l).
-Proof. intros P P' HP [[]]; rewrite /barrier_inv //=. by setoid_rewrite HP. Qed.
+  Proper (dist n ==> eq ==> dist n) (barrier_inv l).
+Proof. solve_proper. Qed.
 Global Instance barrier_ctx_ne n γ l : Proper (dist n ==> dist n) (barrier_ctx γ l).
-Proof. intros P P' HP. rewrite /barrier_ctx. by setoid_rewrite HP. Qed.
+Proof. solve_proper. Qed.
 Global Instance send_ne n l : Proper (dist n ==> dist n) (send l).
-Proof. intros P P' HP. rewrite /send. by setoid_rewrite HP. Qed.
+Proof. solve_proper. Qed.
 Global Instance recv_ne n l : Proper (dist n ==> dist n) (recv l).
-Proof. intros R R' HR. rewrite /recv. by setoid_rewrite HR. Qed.
+Proof. solve_proper. Qed.
 
 (** Helper lemmas *)
 Lemma waiting_split i i1 i2 Q R1 R2 P I :

prelude/tactics.v

@@ -228,6 +228,74 @@ Ltac setoid_subst :=
   | H : @equiv ?A ?e _ ?x |- _ => symmetry in H; setoid_subst_aux (@equiv A e) x
   end.
 
+
+(** f_equiv solves goals of the form "f _ = f _", for any relation and any
+    number of arguments, by looking for appropriate "Proper" instances.
+    If it cannot solve an equality, it will leave that to the user. *)
+Ltac f_equiv :=
+  (* Deal with "pointwise_relation" *)
+  try lazymatch goal with
+    | |- pointwise_relation _ _ _ _ => intros ?
+    end;
+  (* repeatedly apply congruence lemmas and use the equalities in the hypotheses. *)
+  first [ reflexivity | assumption | symmetry; assumption |
+          match goal with
+          (* We support matches on both sides, *if* they concern the same
+             or provably equal variables.
+             TODO: We should support different variables, provided that we can
+             derive contradictions for the off-diagonal cases. *)
+          | |- ?R (match ?x with _ => _ end) (match ?x with _ => _ end) =>
+            destruct x; f_equiv
+          | |- ?R (match ?x with _ => _ end) (match ?y with _ => _ end) =>
+            subst y; f_equiv
+          (* First assume that the arguments need the same relation as the result *)
+          | |- ?R (?f ?x) (?f _) =>
+            let H := fresh "Proper" in
+            assert (Proper (R ==> R) f) as H by (eapply _);
+            apply H; clear H; f_equiv
+          | |- ?R (?f ?x ?y) (?f _ _) =>
+            let H := fresh "Proper" in
+            assert (Proper (R ==> R ==> R) f) as H by (eapply _);
+            apply H; clear H; f_equiv
+          (* Next, try to infer the relation *)
+          (* TODO: If some of the arguments are the same, we could also
+             query for "pointwise_relation"'s. But that leads to a combinatorial
+             explosion about which arguments are and which are not the same. *)
+          | |- ?R (?f ?x) (?f _) =>
+            let R1 := fresh "R" in let H := fresh "Proper" in
+            let T := type of x in evar (R1: relation T);
+            assert (Proper (R1 ==> R) f) as H by (subst R1; eapply _);
+            subst R1; apply H; clear H; f_equiv
+          | |- ?R (?f ?x ?y) (?f _ _) =>
+            let R1 := fresh "R" in let R2 := fresh "R" in
+            let H := fresh "Proper" in
+            let T1 := type of x in evar (R1: relation T1);
+            let T2 := type of y in evar (R2: relation T2);
+            assert (Proper (R1 ==> R2 ==> R) f) as H by (subst R1 R2; eapply _);
+            subst R1 R2; apply H; clear H; f_equiv
+         end | idtac (* Let the user solve this goal *)
+        ].
+
+(** solve_proper solves goals of the form "Proper (R1 ==> R2)", for any
+    number of relations. All the actual work is done by f_equiv;
+    solve_proper just introduces the assumptions and unfolds the first
+    head symbol. *)
+Ltac solve_proper :=
+  (* Introduce everything *)
+  intros;
+  repeat lazymatch goal with
+         | |- Proper _ _ => intros ???
+         | |- (_ ==> _)%signature _ _ => intros ???
+         end;
+  (* Unfold the head symbol, which is the one we are proving a new property about *)
+  lazymatch goal with
+  | |- ?R (?f _ _ _ _) (?f _ _ _ _) => unfold f
+  | |- ?R (?f _ _ _) (?f _ _ _) => unfold f
+  | |- ?R (?f _ _) (?f _ _) => unfold f
+  | |- ?R (?f _) (?f _) => unfold f
+  end;
+  solve [ f_equiv ].
+
 (** Given a tactic [tac2] generating a list of terms, [iter tac1 tac2]
 runs [tac x] for each element [x] until [tac x] succeeds. If it does not
 suceed for any element of the generated list, the whole tactic wil fail. *)

program_logic/saved_prop.v

@@ -43,6 +43,6 @@ Section saved_prop.
     rewrite agree_equivI later_equivI /=; apply later_mono.
     rewrite -{2}(iProp_fold_unfold P) -{2}(iProp_fold_unfold Q).
     apply (eq_rewrite (iProp_unfold P) (iProp_unfold Q) (λ Q' : iPreProp Λ _,
-      iProp_fold (iProp_unfold P) ≡ iProp_fold Q')%I); solve_ne || auto with I.
+      iProp_fold (iProp_unfold P) ≡ iProp_fold Q')%I); solve_proper || auto with I.
   Qed.
 End saved_prop.