let n be Element of NAT ; for x, y being FinSeqLen of n
for a, b being set holds
( (n + 1) -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterStr (x,y)) +* (BitSubtracterWithBorrowStr (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterCirc (x,y)) +* (BitSubtracterWithBorrowCirc (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = BorrowOutput (a,b,(n -BitBorrowOutput (x,y))) )
set c = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)];
let x, y be FinSeqLen of n; for a, b being set holds
( (n + 1) -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterStr (x,y)) +* (BitSubtracterWithBorrowStr (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterCirc (x,y)) +* (BitSubtracterWithBorrowCirc (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = BorrowOutput (a,b,(n -BitBorrowOutput (x,y))) )
let a, b be set ; ( (n + 1) -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterStr (x,y)) +* (BitSubtracterWithBorrowStr (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterCirc (x,y)) +* (BitSubtracterWithBorrowCirc (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = BorrowOutput (a,b,(n -BitBorrowOutput (x,y))) )
set p = x ^ <*a*>;
set q = y ^ <*b*>;
consider f, g, h being ManySortedSet of NAT such that
A1:
n -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = f . n
and
A2:
n -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = 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 of S
for z being set st S = f . n & A = g . n & z = h . n holds
( f . (n + 1) = S +* (BitSubtracterWithBorrowStr (((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z)) & g . (n + 1) = A +* (BitSubtracterWithBorrowCirc (((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z)) & h . (n + 1) = BorrowOutput (((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z) )
by Def2;
A7:
n -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = h . n
by A3, A4, A5, A6, Th1;
A8:
(n + 1) -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = f . (n + 1)
by A3, A4, A5, A6, Th1;
A9:
(n + 1) -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = g . (n + 1)
by A3, A4, A5, A6, Th1;
A10:
(n + 1) -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = h . (n + 1)
by A3, A4, A5, A6, Th1;
A11:
len x = n
by FINSEQ_1:def 18;
A12:
len y = n
by FINSEQ_1:def 18;
A13:
(x ^ <*a*>) . (n + 1) = a
by A11, FINSEQ_1:59;
A14:
(y ^ <*b*>) . (n + 1) = b
by A12, FINSEQ_1:59;
A15:
x ^ <*> = x
by FINSEQ_1:47;
A16:
y ^ <*> = y
by FINSEQ_1:47;
then A17:
n -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = n -BitSubtracterStr (x,y)
by A15, Th5;
A18:
n -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = n -BitSubtracterCirc (x,y)
by A15, A16, Th5;
n -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = n -BitBorrowOutput (x,y)
by A15, A16, Th5;
hence
( (n + 1) -BitSubtracterStr ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterStr (x,y)) +* (BitSubtracterWithBorrowStr (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitSubtracterCirc ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitSubtracterCirc (x,y)) +* (BitSubtracterWithBorrowCirc (a,b,(n -BitBorrowOutput (x,y)))) & (n + 1) -BitBorrowOutput ((x ^ <*a*>),(y ^ <*b*>)) = BorrowOutput (a,b,(n -BitBorrowOutput (x,y))) )
by A1, A2, A6, A7, A8, A9, A10, A13, A14, A17, A18; verum