:: deftheorem Def4 defines -BitAdderCirc FACIRC_2:def 4 :
for n being Nat
for x, y being FinSequence
for b₄ being strict gate`2=den Boolean Circuit of n -BitAdderStr (x,y) holds
( b₄ = n -BitAdderCirc (x,y) iff ex f, g, h being ManySortedSet of NAT st
( n -BitAdderStr (x,y) = f . n & b₄ = g . n & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for z being set st S = f . n & A = g . n & z = h . n holds
( f . (n + 1) = S +* (BitAdderWithOverflowStr ((x . (n + 1)),(y . (n + 1)),z)) & g . (n + 1) = A +* (BitAdderWithOverflowCirc ((x . (n + 1)),(y . (n + 1)),z)) & h . (n + 1) = MajorityOutput ((x . (n + 1)),(y . (n + 1)),z) ) ) ) );