simple induction b.
simple induction c.
auto.
unfold full_adder_sum in |- *.
rewrite half_adder_sum_false.
rewrite half_adder_sum_false.
auto.
unfold full_adder_sum in |- *.
rewrite half_adder_sum_false.
intro.
auto.
Qed. Hint Resolve full_adder_sum_sym2.

Lemma full_adder_sum_false :
forall a : bool, full_adder_sum a false false = a.