diff --git a/quoting.v b/quoting.v
index 60e083b..b2976f1 100644
--- a/quoting.v
+++ b/quoting.v
@@ -77,24 +77,14 @@ Fixpoint pat_ind'
          p
          : P p.
 Proof.
-  induction p; auto.
-
-  apply PRecord.
-  apply IHp; try assumption.
-  induction l; [ apply Forall_nil | ].
-  apply Forall_cons; [ apply pat_ind'; try assumption | ].
-  apply IHl; apply IHp.
-
-  apply PSeq.
-  induction l; [ apply Forall_nil | ].
-  apply Forall_cons; [ apply pat_ind'; try assumption | ].
-  apply IHl.
+  induction p; auto; [ apply PRecord; [ apply IHp | ] | apply PSeq ];
+    try solve [ induction l; [ apply Forall_nil | ];
+                apply Forall_cons; [ apply pat_ind'; try assumption | ];
+                apply IHl ].
 Defined.
 
-Lemma Forall_map_map_id :
-  forall ps,
-    Forall (fun p : pat => uqp (qp p) = p) ps ->
-    (map uqp (map qp ps)) = ps.
+Lemma Forall_map_map_id : forall ps, Forall (fun p : pat => uqp (qp p) = p) ps ->
+                                     (map uqp (map qp ps)) = ps.
 Proof.
   intros; rewrite map_map; induction H; [ reflexivity | ].
   simpl; rewrite IHForall; rewrite H; reflexivity.