(** Test if, starting at position [n], [s1] occurs in [s2]; if so it returns the position *)
Fixpoint index (n : nat) (s1 s2 : string) : option nat :=
match s2, n with
| EmptyString, O =>
match s1 with
| EmptyString => Some O
| String a s1' => None
end
| EmptyString, S n' => None
| String b s2', O =>
if prefix s1 s2 then Some O
else
match index O s1 s2' with
| Some n => Some (S n)
| None => None
end
| String b s2', S n' =>
match index n' s1 s2' with
| Some n => Some (S n)
| None => None
end
end.
(* Dirty trick to avoid locally that prefix reduces itself *)
Opaque prefix.
(** If the result of [index] is [Some m], [s1] in [s2] at position [m] *)

Theorem index_correct1 :
forall (n m : nat) (s1 s2 : string),
index n s1 s2 = Some m -> substring m (length s1) s2 = s1.