let n be Element of NAT ; 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; ( (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 NAT such that
A1:
n -BitAdderStr x,y = f . n
and
A2:
n -BitAdderCirc x,y = g . n
and
A3:
f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE )
and
A4:
g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE )
and
A5:
h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )]
and
A6:
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;
A7:
n -BitMajorityOutput x,y = h . n
by A3, A4, A5, A6, Th7;
A8:
(n + 1) -BitAdderStr x,y = f . (n + 1)
by A3, A4, A5, A6, Th7;
A9:
(n + 1) -BitAdderCirc x,y = g . (n + 1)
by A3, A4, A5, A6, Th7;
(n + 1) -BitMajorityOutput x,y = h . (n + 1)
by A3, A4, A5, A6, 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, A2, A6, A7, A8, A9; verum