Some cleaning in list representation

FiniteSets/kuratowski/length.v

@@ -11,15 +11,17 @@ Section length.
       apply (if a ∈_d X then n else (S n)).
     - intros X a n.
       simpl.
-      simplify_isIn_d.
-      destruct (dec (a ∈ X)) ; reflexivity.
+      rewrite ?union_isIn_d, singleton_isIn_d_aa.
+      reflexivity.
     - intros X a b n.
       simpl.
-      simplify_isIn_d.
+      rewrite ?union_isIn_d.
       destruct (m_dec_path a b) as [Hab | Hab].
       + strip_truncations.
-        rewrite Hab. simplify_isIn_d. reflexivity.
-      + rewrite ?singleton_isIn_d_false; auto.
+        rewrite Hab.
+        rewrite ?singleton_isIn_d_aa.
+        reflexivity.
+      + rewrite ?singleton_isIn_d_false.
         ++ simpl. 
            destruct (a ∈_d X), (b ∈_d X) ; reflexivity.
         ++ intro p. contradiction (Hab (tr p^)).

FiniteSets/kuratowski/operations.v

@@ -219,18 +219,12 @@ Section FSet_cons_rec.
            (Pcomm : forall X a b p, Pcons a ({|b|} ∪ X) (Pcons b X p)
                                            = Pcons b ({|a|} ∪ X) (Pcons a X p)).
   
-  Definition FSet_cons_rec (X : FSet A) : P.
-  Proof.
-    simple refine (FSetC_ind A (fun _ => P) _ Pe _ _ _ (FSet_to_FSetC X)) ; simpl.
-    - intros a Y p.
-      apply (Pcons a (FSetC_to_FSet Y) p).
-    - intros.
-      refine (transport_const _ _ @ _).
-      apply Pdupl.
-    - intros.
-      refine (transport_const _ _ @ _).
-      apply Pcomm.
-  Defined.
+  Definition FSet_cons_rec (X : FSet A) : P :=
+    FSetC_prim_rec A P Pset Pe
+                   (fun a Y p => Pcons a (FSetC_to_FSet Y) p)
+                   (fun _ _ => Pdupl _ _)
+                   (fun _ _ _ => Pcomm _ _ _)
+                   (FSet_to_FSetC X).
 
   Definition FSet_cons_beta_empty : FSet_cons_rec ∅ = Pe := idpath.