improve f_equiv and solve_proper; use them in a few more places

theories/tactics.v

@@ -257,22 +257,28 @@ Ltac f_equiv :=
             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 *)
+          (* Next, try to infer the relation. Unfortunately, there is an instance
+             of Proper for (eq ==> _), which will always be matched. *)
+          (* TODO: Can we exclude that instance? *)
           (* 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 R1 := fresh "R" in let H := fresh "HProp" 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 H := fresh "HProp" 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
+           (* In case the function symbol differs, but the arguments are the same,
+              maybe we have a pointwise_relation in our context. *)
+           | H : pointwise_relation _ ?R ?f ?g |- ?R (?f ?x) (?g ?x) =>
+             apply H; f_equiv
          end | idtac (* Let the user solve this goal *)
         ].
 
@@ -289,6 +295,10 @@ Ltac solve_proper :=
          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
   | |- ?R (?f _ _ _ _) (?f _ _ _ _) => unfold f
   | |- ?R (?f _ _ _) (?f _ _ _) => unfold f
   | |- ?R (?f _ _) (?f _ _) => unfold f
@@ -296,6 +306,7 @@ Ltac solve_proper :=
   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. *)