let n be Element of NAT ; :: thesis: for x, y being FinSequence holds
( (n + 1) -BitSubtracterStr x,y = (n -BitSubtracterStr x,y) +* (BitSubtracterWithBorrowStr (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y)) & (n + 1) -BitSubtracterCirc x,y = (n -BitSubtracterCirc x,y) +* (BitSubtracterWithBorrowCirc (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y)) & (n + 1) -BitBorrowOutput x,y = BorrowOutput (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y) )
let x, y be FinSequence; :: thesis: ( (n + 1) -BitSubtracterStr x,y = (n -BitSubtracterStr x,y) +* (BitSubtracterWithBorrowStr (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y)) & (n + 1) -BitSubtracterCirc x,y = (n -BitSubtracterCirc x,y) +* (BitSubtracterWithBorrowCirc (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y)) & (n + 1) -BitBorrowOutput x,y = BorrowOutput (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y) )
set c = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )];
consider f, g, h being ManySortedSet of such that
A1: ( n -BitSubtracterStr x,y = f . n & n -BitSubtracterCirc x,y = g . n ) and
A2: ( f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] ) 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 +* (BitSubtracterWithBorrowStr (x . (n + 1)),(y . (n + 1)),z) & g . (n + 1) = A +* (BitSubtracterWithBorrowCirc (x . (n + 1)),(y . (n + 1)),z) & h . (n + 1) = BorrowOutput (x . (n + 1)),(y . (n + 1)),z ) by Def2;
( n -BitBorrowOutput x,y = h . n & (n + 1) -BitSubtracterStr x,y = f . (n + 1) & (n + 1) -BitSubtracterCirc x,y = g . (n + 1) & (n + 1) -BitBorrowOutput x,y = h . (n + 1) ) by A2, A3, Th1;
hence ( (n + 1) -BitSubtracterStr x,y = (n -BitSubtracterStr x,y) +* (BitSubtracterWithBorrowStr (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y)) & (n + 1) -BitSubtracterCirc x,y = (n -BitSubtracterCirc x,y) +* (BitSubtracterWithBorrowCirc (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y)) & (n + 1) -BitBorrowOutput x,y = BorrowOutput (x . (n + 1)),(y . (n + 1)),(n -BitBorrowOutput x,y) ) by A1, A3; :: thesis: verum