simple induction b.
simple induction c.
auto.
unfold full_adder_carry in |- *.
rewrite half_adder_sum_false.
rewrite half_adder_carry_false.
rewrite half_adder_carry_false.
simpl in |- *.
elim (half_adder_carry a true); auto.
intros.
unfold full_adder_carry in |- *.
rewrite half_adder_carry_false.
rewrite half_adder_sum_false.
rewrite half_adder_carry_false.
simpl in |- *.
elim (half_adder_carry a c); auto.
Qed. Hint Resolve full_adder_carry_sym2.

Lemma full_adder_carry_false :
forall a : bool, full_adder_carry a false false = false.