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 member_case
: forall x ls (m : member x ls),
match ls as ls return member x ls -> Prop with
| nil => fun _ => False
| l :: ls' => fun m =>
(exists (pf : l = x),
m = match pf in _ = z return member z (l :: ls') with
| eq_refl => MZ _ ls'
end)
\/ exists m' : member x ls',
m = MN _ m'
end m.