let a, b be FinSequence; :: thesis: ( 0 -BitGFA0Str a,b = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & 0 -BitGFA0Circ a,b = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & 0 -BitGFA0CarryOutput a,b = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] )
set f0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE );
set g0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE );
set h0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )];
consider f, g, h being ManySortedSet of such that
A1: ( 0 -BitGFA0Str a,b = f . 0 & 0 -BitGFA0Circ a,b = g . 0 ) and
A2: f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) and
A3: g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) and
h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] and
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 +* (BitGFA0Str (a . (n + 1)),(b . (n + 1)),z) & g . (n + 1) = A +* (BitGFA0Circ (a . (n + 1)),(b . (n + 1)),z) & h . (n + 1) = GFA0CarryOutput (a . (n + 1)),(b . (n + 1)),z ) by Def2;
thus 0 -BitGFA0Str a,b = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) by A1, A2; :: thesis: ( 0 -BitGFA0Circ a,b = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & 0 -BitGFA0CarryOutput a,b = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] )
thus 0 -BitGFA0Circ a,b = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) by A1, A3; :: thesis: 0 -BitGFA0CarryOutput a,b = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )]
InnerVertices (0 -BitGFA0Str a,b) = {[<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )]} by A1, A2, CIRCCOMB:49;
hence 0 -BitGFA0CarryOutput a,b = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] by TARSKI:def 1; :: thesis: verum