Inductive bin:=
   OO: bin
  | SS: bin -> bin
  | TT: bin -> bin.Inductive bin:=
   OO: bin
  | SS: bin -> bin
  | TT: bin -> bin.Inductive bin:=
   OO: bin
  | SS: bin -> bin
  | TT: bin -> bin.Inductive bin:=
   OO: bin
  | SS: bin -> bin
  | TT: bin -> bin.

Theorem bin_to_nat_pres_incr : forall b : bin, bin_to_nat (incr b) = plus (bin_to_nat b) 1.