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; :: thesis: 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 ; :: thesis: ( 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 )] ; :: thesis: ( 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 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 ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = c ) and
A5: 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;
( c in the carrier of (BitGFA1Str (a . 1),(b . 1),c) & 1 -BitGFA1CarryOutput a,b = h . (0 + 1) ) by A1, A4, A5, Th15, GFACIRC1:81;
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; :: thesis: verum