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 of 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