set f0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE));
set g0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE));
set h0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)];
let n be Nat; for x, y being FinSeqLen of n
for a, b being set holds
( (n + 1) -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Str (x,y)) +* (BitGFA0Str (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Circ (x,y)) +* (BitGFA0Circ (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = GFA0CarryOutput (a,b,(n -BitGFA0CarryOutput (x,y))) )
let x, y be FinSeqLen of n; for a, b being set holds
( (n + 1) -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Str (x,y)) +* (BitGFA0Str (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Circ (x,y)) +* (BitGFA0Circ (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = GFA0CarryOutput (a,b,(n -BitGFA0CarryOutput (x,y))) )
let a, b be set ; ( (n + 1) -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Str (x,y)) +* (BitGFA0Str (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Circ (x,y)) +* (BitGFA0Circ (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = GFA0CarryOutput (a,b,(n -BitGFA0CarryOutput (x,y))) )
set p = x ^ <*a*>;
set q = y ^ <*b*>;
consider f, g, h being ManySortedSet of NAT such that
A1:
n -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = f . n
and
A2:
n -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = 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 over S
for z being set st S = f . n & A = g . n & z = h . n holds
( f . (n + 1) = S +* (BitGFA0Str (((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z)) & g . (n + 1) = A +* (BitGFA0Circ (((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z)) & h . (n + 1) = GFA0CarryOutput (((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z) )
by Def2;
A7:
n -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = h . n
by A3, A4, A5, A6, Th1;
A8:
(n + 1) -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = f . (n + 1)
by A3, A4, A5, A6, Th1;
A9:
(n + 1) -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = g . (n + 1)
by A3, A4, A5, A6, Th1;
A10:
(n + 1) -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = h . (n + 1)
by A3, A4, A5, A6, Th1;
A11:
len x = n
by CARD_1:def 7;
A12:
len y = n
by CARD_1:def 7;
A13:
(x ^ <*a*>) . (n + 1) = a
by A11, FINSEQ_1:42;
A14:
(y ^ <*b*>) . (n + 1) = b
by A12, FINSEQ_1:42;
A15:
x ^ <*> = x
by FINSEQ_1:34;
A16:
y ^ <*> = y
by FINSEQ_1:34;
then A17:
n -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = n -BitGFA0Str (x,y)
by A15, Th5;
A18:
n -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = n -BitGFA0Circ (x,y)
by A15, A16, Th5;
n -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = n -BitGFA0CarryOutput (x,y)
by A15, A16, Th5;
hence
( (n + 1) -BitGFA0Str ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Str (x,y)) +* (BitGFA0Str (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0Circ ((x ^ <*a*>),(y ^ <*b*>)) = (n -BitGFA0Circ (x,y)) +* (BitGFA0Circ (a,b,(n -BitGFA0CarryOutput (x,y)))) & (n + 1) -BitGFA0CarryOutput ((x ^ <*a*>),(y ^ <*b*>)) = GFA0CarryOutput (a,b,(n -BitGFA0CarryOutput (x,y))) )
by A1, A2, A6, A7, A8, A9, A10, A13, A14, A17, A18; verum