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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    *)
(*                                                              *)
(* file: HalfAdder.v                                            *)
(* contents: definition and verification of an half-adder       *)
(*                                                              *)
(****************************************************************)
(* Half adder sur les bits (bool) : rend la somme ou le carry *)
(* Verification: Theorem half_adder_ok *)
Require Export Arith_compl.
Require Export Bool_compl.
(****************************************************************)
Definition half_adder_sum (a b : bool) := xorb a b.
Definition half_adder_carry (a b : bool) := a && b.
(****************************************************************)
simple induction a; simple induction b; auto.
Qed. Hint Resolve half_adder_sum_sym.
simple induction a; simple induction b; auto.
Qed. Hint Resolve half_adder_carry_sym.
simple induction a; auto.
Qed. Hint Resolve half_adder_sum_false.
simple induction a; auto.
Qed. Hint Resolve half_adder_carry_false.
auto.
Qed. Hint Resolve half_adder_sum_true.
simple induction a; auto.
Qed. Hint Resolve half_adder_carry_true.
(****************************************************************)

Theorem half_adder_ok :
forall a b : bool,
bool_to_nat (half_adder_sum a b) +
(bool_to_nat (half_adder_carry a b) + bool_to_nat (half_adder_carry a b)) =
bool_to_nat a + bool_to_nat b.