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)];
A1: ex f, g, h being ManySortedSet of NAT st
( 0 -BitGFA0Str (a,b) = f . 0 & 0 -BitGFA0Circ (a,b) = g . 0 & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] & ( 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 +* (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;
hence 0 -BitGFA0Str (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE)) ; :: 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; :: thesis: 0 -BitGFA0CarryOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)]
InnerVertices (0 -BitGFA0Str (a,b)) = {[<*>,((0 -tuples_on BOOLEAN) --> FALSE)]} by A1, CIRCCOMB:42;
hence 0 -BitGFA0CarryOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)] by TARSKI:def 1; :: thesis: verum