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 ap, bp, cp, dp being FinSequence holds
( (n + 1) -BitFTA0Str (ap,bp,cp,dp) = (n -BitFTA0Str (ap,bp,cp,dp)) +* (BitFTA0Str ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp)))) & (n + 1) -BitFTA0Circ (ap,bp,cp,dp) = (n -BitFTA0Circ (ap,bp,cp,dp)) +* (BitFTA0Circ ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp)))) & (n + 1) -BitFTA0CarryOutput (ap,bp,cp,dp) = BitFTA0CarryOutput ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp))) )
let ap, bp, cp, dp be FinSequence; ( (n + 1) -BitFTA0Str (ap,bp,cp,dp) = (n -BitFTA0Str (ap,bp,cp,dp)) +* (BitFTA0Str ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp)))) & (n + 1) -BitFTA0Circ (ap,bp,cp,dp) = (n -BitFTA0Circ (ap,bp,cp,dp)) +* (BitFTA0Circ ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp)))) & (n + 1) -BitFTA0CarryOutput (ap,bp,cp,dp) = BitFTA0CarryOutput ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp))) )
consider f, g, h being ManySortedSet of NAT such that
A1:
n -BitFTA0Str (ap,bp,cp,dp) = f . n
and
A2:
n -BitFTA0Circ (ap,bp,cp,dp) = 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 +* (BitFTA0Str ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),z)) & g . (n + 1) = A +* (BitFTA0Circ ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),z)) & h . (n + 1) = BitFTA0CarryOutput ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),z) )
by Def2;
A7:
n -BitFTA0CarryOutput (ap,bp,cp,dp) = h . n
by A3, A4, A5, A6, Th1;
A8:
(n + 1) -BitFTA0Str (ap,bp,cp,dp) = f . (n + 1)
by A3, A4, A5, A6, Th1;
A9:
(n + 1) -BitFTA0Circ (ap,bp,cp,dp) = g . (n + 1)
by A3, A4, A5, A6, Th1;
(n + 1) -BitFTA0CarryOutput (ap,bp,cp,dp) = h . (n + 1)
by A3, A4, A5, A6, Th1;
hence
( (n + 1) -BitFTA0Str (ap,bp,cp,dp) = (n -BitFTA0Str (ap,bp,cp,dp)) +* (BitFTA0Str ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp)))) & (n + 1) -BitFTA0Circ (ap,bp,cp,dp) = (n -BitFTA0Circ (ap,bp,cp,dp)) +* (BitFTA0Circ ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp)))) & (n + 1) -BitFTA0CarryOutput (ap,bp,cp,dp) = BitFTA0CarryOutput ((ap . (n + 1)),(bp . (n + 1)),(cp . (n + 1)),(dp . (n + 1)),(n -BitFTA0CarryOutput (ap,bp,cp,dp))) )
by A1, A2, A6, A7, A8, A9; verum