assert (HH: Is_S O = False) by reflexivity.
rewrite <- HH.
rewrite H. simpl.
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.