set f0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set g0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set h0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )];
let n be Nat; :: thesis: for x, y being FinSequence holds
( (n + 1) -BitGFA1Str x,y = (n -BitGFA1Str x,y) +* (BitGFA1Str (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ x,y = (n -BitGFA1Circ x,y) +* (BitGFA1Circ (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput x,y = GFA1CarryOutput (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y) )
let x, y be FinSequence; :: thesis: ( (n + 1) -BitGFA1Str x,y = (n -BitGFA1Str x,y) +* (BitGFA1Str (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ x,y = (n -BitGFA1Circ x,y) +* (BitGFA1Circ (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput x,y = GFA1CarryOutput (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y) )
consider f, g, h being ManySortedSet of such that
A1: ( n -BitGFA1Str x,y = f . n & n -BitGFA1Circ x,y = g . n ) and
A2: ( f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] ) and
A3: 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 +* (BitGFA1Str (x . (n + 1)),(y . (n + 1)),z) & g . (n + 1) = A +* (BitGFA1Circ (x . (n + 1)),(y . (n + 1)),z) & h . (n + 1) = GFA1CarryOutput (x . (n + 1)),(y . (n + 1)),z ) by Def6;
( n -BitGFA1CarryOutput x,y = h . n & (n + 1) -BitGFA1Str x,y = f . (n + 1) & (n + 1) -BitGFA1Circ x,y = g . (n + 1) & (n + 1) -BitGFA1CarryOutput x,y = h . (n + 1) ) by A2, A3, Th15;
hence ( (n + 1) -BitGFA1Str x,y = (n -BitGFA1Str x,y) +* (BitGFA1Str (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ x,y = (n -BitGFA1Circ x,y) +* (BitGFA1Circ (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput x,y = GFA1CarryOutput (x . (n + 1)),(y . (n + 1)),(n -BitGFA1CarryOutput x,y) ) by A1, A3; :: thesis: verum