:: deftheorem Def3 defines -BitAdderStr FACIRC_2:def 3 :
for n being Nat
for x, y being FinSequence
for b₄ being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign holds
( b₄ = n -BitAdderStr (x,y) iff ex f, g being ManySortedSet of NAT st
( b₄ = f . n & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & g . 0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] & ( for n being Nat
for S being non empty ManySortedSign
for z being set st S = f . n & z = g . n holds
( f . (n + 1) = S +* (BitAdderWithOverflowStr ((x . (n + 1)),(y . (n + 1)),z)) & g . (n + 1) = MajorityOutput ((x . (n + 1)),(y . (n + 1)),z) ) ) ) );