(* ** 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) [].
(* strings of natural numbers to bitstrings, [ n1, ... n2 ] |-> 1^n1 0 ... 1^n2 0 *)
Fixpoint f_s (x : string nat) : string bool :=
match x with
| nil => nil
| a :: x => to_bitstring a ++ [false] ++ f_s x
end.
(* extension to cards and stacks *)
Definition f_c '(x,y) := (f_s x, f_s y).
Definition f (P : stack nat) : stack bool :=
map f_c P.

Lemma tau2_f A : tau2 (f A) = f_s (tau2 A).