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.
Fixpoint nth_member (ls : list T) (n : nat) {struct n}
: option { x : T & member x ls } :=
match ls as ls return option { x : T & member x ls } with
| nil => None
| l :: ls =>
match n with
| 0 => Some (@existT _ (fun x => member x (l :: ls)) l (MZ _ _))
| S n =>
match nth_member ls n with
| Some (existT v m) => Some (@existT _ _ v (MN _ m))
| None => None
end
end
end.
Definition get_next ls y x (m : member x (y :: ls))
: option (member x ls) :=
match m in member _ ls'
return match ls' with
| nil => unit
| l' :: ls' => option (member x ls')
end
with
| MZ _ => None
| MN _ _ m => Some m
end.
Instance Injective_MN x y ls m m' : Injective (@MN x y ls m = @MN x y ls m').

Lemma member_In : forall ls (t : T), member t ls -> List.In t ls.