set S0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set A0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set Sn = n -BitGFA1Str (x,y);
set o0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)];
deffunc H₁( non empty ManySortedSign , set , Nat) -> ManySortedSign = $1 +* (BitGFA1Str ((x . ($3 + 1)),(y . ($3 + 1)),$2));
deffunc H₂( non empty ManySortedSign , non-empty MSAlgebra over $1, set , Nat) -> MSAlgebra over $1 +* (BitGFA1Str ((x . ($4 + 1)),(y . ($4 + 1)),$3)) = $2 +* (BitGFA1Circ ((x . ($4 + 1)),(y . ($4 + 1)),$3));
deffunc H₃( set , Nat) -> Element of InnerVertices (GFA1CarryStr ((x . ($2 + 1)),(y . ($2 + 1)),$1)) = GFA1CarryOutput ((x . ($2 + 1)),(y . ($2 + 1)),$1);
A2: for S being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign
for x being set
for n being Nat holds
( H₁(S,x,n) is unsplit & H₁(S,x,n) is gate`1=arity & H<sub>1</sub>(S,x,n) is gate`2isBoolean & not H₁(S,x,n) is void & H₁(S,x,n) is strict ) ;
A3: ex f, h being ManySortedSet of NAT st
( n -BitGFA1Str (x,y) = f . n & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] & ( for n being Nat
for S being non empty ManySortedSign
for x being set st S = f . n & x = h . n holds
( f . (n + 1) = H₁(S,x,n) & h . (n + 1) = H₃(x,n) ) ) ) by Def5;
A4: for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set
for n being Nat holds H₂(S,A,x,n) is non-empty MSAlgebra over H₁(S,x,n) ;
A5: for S, S1 being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign
for A being strict gate`2=den Boolean Circuit of S
for x being set
for n being Nat st S1 = H<sub>1</sub>(S,x,n) holds
H<sub>2</sub>(S,A,x,n) is strict gate`2=den Boolean Circuit of S1 ;
ex A being strict gate`2=den Boolean Circuit of n -BitGFA1Str (x,y) ex f, g, h being ManySortedSet of NAT st
( n -BitGFA1Str (x,y) = f . n & A = g . n & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = H<sub>1</sub>(S,x,n) & g . (n + 1) = H<sub>2</sub>(S,A,x,n) & h . (n + 1) = H<sub>3</sub>(x,n) ) ) ) from CIRCCMB2:sch 19(A2, A3, A4, A5);
hence ex b<sub>1</sub> being strict gate`2=den Boolean Circuit of n -BitGFA1Str (x,y) ex f, g, h being ManySortedSet of NAT st
( n -BitGFA1Str (x,y) = f . n & b₁ = g . n & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] & ( 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 +* (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) ) ) ) ; :: thesis: verum