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Require Import TestSuite.admit.
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 _ _ _ _ ?[def]).
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).
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