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+Section Counted_list.
+
+  Variable A : Type.
+
+  Inductive counted_list : nat -> Type :=
+    | counted_nil : counted_list 0
+    | counted_cons : forall(n : nat), 
+        A -> counted_list n -> counted_list (S n).
+
+
+  Fixpoint counted_def_nth{n : nat}(l : counted_list n)
+                          (i : nat)(def : A) : A :=
+    match i with
+      | 0 => match l with
+               | counted_nil => def
+               | counted_cons _ a _ => a
+             end
+      | S i => match l with
+                 | counted_nil => def
+                 | counted_cons _ _ tl => counted_def_nth tl i def
+               end
+    end.
+
+
+  Lemma counted_list_equal_nth_char : 
+    forall(n : nat)(l1 l2 : counted_list n)(def : A),
+      (forall(i : nat), counted_def_nth l1 i def = counted_def_nth l2 i def) ->
+        l1 = l2.
+  Proof.
+    admit.
+  Qed.
+
+End Counted_list.
+
+Implicit Arguments counted_def_nth [A n].
+
+Section Finite_nat_set.
+
+  Variable set_size : nat.
+
+  Definition fnat_subset : Type := counted_list bool set_size.
+
+  Definition fnat_member(fs : fnat_subset)(n : nat) : Prop := 
+    is_true (counted_def_nth fs n false).
+
+
+  Lemma fnat_subset_member_eq : forall(fs1 fs2 : fnat_subset),
+    fs1 = fs2 <->
+      forall(n : nat), fnat_member fs1 n <-> fnat_member fs2 n.
+
+  Proof.
+    intros fs1 fs2.
+    split.
+      intros H n.
+      subst fs1.
+      apply iff_refl.
+    intros H.
+    eapply counted_list_equal_nth_char.
+    intros i.
+    destruct (counted_def_nth fs1 i _ ) eqn:H0.
+    (* This was not part of the initial bug report; this is to check that
+       the existential variable kept its name *)
+    change (true = counted_def_nth fs2 i ?def).