Section to_nat.
Variable a : T.
Fixpoint to_nat {ls} (m : member a ls) : nat :=
match m with
| MZ _ => 0
| MN _ _ m => S (to_nat m)
end.
End to_nat.

Lemma to_nat_eq_member_eq
: forall {_ : EqDec T (@eq T)} x ls (a b : member x ls),
to_nat a = to_nat b ->
a = b.