let n be Nat; 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; ( (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 NAT such that
A1:
n -BitSubtracterStr (x,y) = f . n
and
A2:
n -BitSubtracterCirc (x,y) = g . n
and
A3:
f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))
and
A4:
g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))
and
A5:
h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)]
and
A6:
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 +* (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;
A7:
n -BitBorrowOutput (x,y) = h . n
by A3, A4, A5, A6, Th1;
A8:
(n + 1) -BitSubtracterStr (x,y) = f . (n + 1)
by A3, A4, A5, A6, Th1;
A9:
(n + 1) -BitSubtracterCirc (x,y) = g . (n + 1)
by A3, A4, A5, A6, Th1;
(n + 1) -BitBorrowOutput (x,y) = h . (n + 1)
by A3, A4, A5, A6, 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, A2, A6, A7, A8, A9; verum