(** ** PCP reduces to BPCP *)
(* natural numbers n to bitstrings of the form 1^n *)
Definition to_bitstring (n : nat) : string bool := Nat.iter n (cons true) [].

Lemma bitstring_false a : ~ false el to_bitstring a.
Fixpoint f_s (x : string nat) : string bool :=
match x with
| nil => nil
| a :: x => to_bitstring a ++ [false] ++ f_s x
end.