let n be Element of NAT ; :: thesis: for x, y being FinSequence holds
( (n + 1) -BitAdderStr x,y = (n -BitAdderStr x,y) +* (BitAdderWithOverflowStr (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y)) & (n + 1) -BitAdderCirc x,y = (n -BitAdderCirc x,y) +* (BitAdderWithOverflowCirc (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y)) & (n + 1) -BitMajorityOutput x,y = MajorityOutput (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y) )
let x, y be FinSequence; :: thesis: ( (n + 1) -BitAdderStr x,y = (n -BitAdderStr x,y) +* (BitAdderWithOverflowStr (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y)) & (n + 1) -BitAdderCirc x,y = (n -BitAdderCirc x,y) +* (BitAdderWithOverflowCirc (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y)) & (n + 1) -BitMajorityOutput x,y = MajorityOutput (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y) )
set c = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )];
consider f, g, h being ManySortedSet of such that
A1: ( n -BitAdderStr x,y = f . n & n -BitAdderCirc x,y = g . n ) and
A2: ( f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] ) and
A3: for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra of 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 ) by Def4;
( n -BitMajorityOutput x,y = h . n & (n + 1) -BitAdderStr x,y = f . (n + 1) & (n + 1) -BitAdderCirc x,y = g . (n + 1) & (n + 1) -BitMajorityOutput x,y = h . (n + 1) ) by A2, A3, Th7;
hence ( (n + 1) -BitAdderStr x,y = (n -BitAdderStr x,y) +* (BitAdderWithOverflowStr (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y)) & (n + 1) -BitAdderCirc x,y = (n -BitAdderCirc x,y) +* (BitAdderWithOverflowCirc (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y)) & (n + 1) -BitMajorityOutput x,y = MajorityOutput (x . (n + 1)),(y . (n + 1)),(n -BitMajorityOutput x,y) ) by A1, A3; :: thesis: verum