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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************)
(*           The Calculus of Inductive Constructions            *)
(*                       COQ v5.10                              *)
(*                                                              *)
(* Laurent Arditi.  Laboratoire I3S. CNRS ura 1376.             *)
(* Universite de Nice - Sophia Antipolis                        *)
(* arditi@unice.fr, http://wwwi3s.unice.fr/~arditi/lolo.html    *)
(*                                                              *)
(* date: may 1995                                               *)
(* file: FullAdder.v                                            *)
(* contents: definition and verification of a full adder        *)
(*                                                              *)
(****************************************************************)
(* Full adder sur les bits (bool) : rend la somme ou le carry *)
(* Verification: Theorem full_adder_ok *)
Require Export HalfAdder.
(****************************************************************)
Definition full_adder_sum (a b c : bool) :=
half_adder_sum (half_adder_sum a b) c.
Definition full_adder_carry (a b c : bool) :=
half_adder_carry a b || half_adder_carry (half_adder_sum a b) c.
(****************************************************************)
simple induction a; simple induction b; auto.
Qed. Hint Resolve full_adder_sum_sym1.
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.
simple induction a; auto.
Qed. Hint Resolve full_adder_sum_false.
simple induction a; auto.
Qed. Hint Resolve full_adder_sum_true.
simple induction a; simple induction b; auto.
Qed. Hint Resolve full_adder_carry_sym1.
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.
simple induction a; auto.
Qed. Hint Resolve full_adder_carry_false.
simple induction a.
unfold full_adder_carry in |- *.
auto.
unfold full_adder_carry in |- *.
auto.
Qed. Hint Resolve full_adder_carry_true.
simple induction a; auto.
Qed. Hint Resolve full_adder_carry_true_false.
simple induction a; simple induction b; simpl in |- *.
rewrite full_adder_carry_sym1. rewrite full_adder_carry_true. trivial.
rewrite full_adder_carry_false. trivial.
rewrite full_adder_carry_true. trivial.
rewrite full_adder_carry_sym1. rewrite full_adder_carry_false. trivial.
Qed.
(****************************************************************)

Theorem full_adder_ok :
forall a b c : bool,
bool_to_nat (full_adder_sum a b c) +
(bool_to_nat (full_adder_carry a b c) + bool_to_nat (full_adder_carry a b c)) =
bool_to_nat a + bool_to_nat b + bool_to_nat c.