set f0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set g0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set h0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)];
let a, b be FinSequence; for c being set st c = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] holds
( 1 -BitGFA1Str (a,b) = (1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Str ((a . 1),(b . 1),c)) & 1 -BitGFA1Circ (a,b) = (1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Circ ((a . 1),(b . 1),c)) & 1 -BitGFA1CarryOutput (a,b) = GFA1CarryOutput ((a . 1),(b . 1),c) )
let c be set ; ( c = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] implies ( 1 -BitGFA1Str (a,b) = (1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Str ((a . 1),(b . 1),c)) & 1 -BitGFA1Circ (a,b) = (1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Circ ((a . 1),(b . 1),c)) & 1 -BitGFA1CarryOutput (a,b) = GFA1CarryOutput ((a . 1),(b . 1),c) ) )
assume A1:
c = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)]
; ( 1 -BitGFA1Str (a,b) = (1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Str ((a . 1),(b . 1),c)) & 1 -BitGFA1Circ (a,b) = (1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Circ ((a . 1),(b . 1),c)) & 1 -BitGFA1CarryOutput (a,b) = GFA1CarryOutput ((a . 1),(b . 1),c) )
consider f, g, h being ManySortedSet of NAT such that
A2:
1 -BitGFA1Str (a,b) = f . 1
and
A3:
1 -BitGFA1Circ (a,b) = g . 1
and
A4:
f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))
and
A5:
g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))
and
A6:
h . 0 = c
and
A7:
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 +* (BitGFA1Str ((a . (n + 1)),(b . (n + 1)),z)) & g . (n + 1) = A +* (BitGFA1Circ ((a . (n + 1)),(b . (n + 1)),z)) & h . (n + 1) = GFA1CarryOutput ((a . (n + 1)),(b . (n + 1)),z) )
by A1, Def6;
1 -BitGFA1CarryOutput (a,b) = h . (0 + 1)
by A1, A4, A5, A6, A7, Th15;
hence
( 1 -BitGFA1Str (a,b) = (1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Str ((a . 1),(b . 1),c)) & 1 -BitGFA1Circ (a,b) = (1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE))) +* (BitGFA1Circ ((a . 1),(b . 1),c)) & 1 -BitGFA1CarryOutput (a,b) = GFA1CarryOutput ((a . 1),(b . 1),c) )
by A2, A3, A4, A5, A6, A7; verum