set S0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> FALSE));
set A0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> FALSE));
set Sn = n -BitGFA0Str (x,y);
set o0 = [<*>,((0 -tuples_on BOOLEAN) --> FALSE)];
deffunc H₁( non empty ManySortedSign , set , Nat) -> ManySortedSign = $1 +* (BitGFA0Str ((x . ($3 + 1)),(y . ($3 + 1)),$2));
deffunc H₂( non empty ManySortedSign , non-empty MSAlgebra over $1, set , Nat) -> MSAlgebra over $1 +* (BitGFA0Str ((x . ($4 + 1)),(y . ($4 + 1)),$3)) = $2 +* (BitGFA0Circ ((x . ($4 + 1)),(y . ($4 + 1)),$3));
deffunc H₃( set , Nat) -> Element of InnerVertices (GFA0CarryStr ((x . ($2 + 1)),(y . ($2 + 1)),$1)) = GFA0CarryOutput ((x . ($2 + 1)),(y . ($2 + 1)),$1);
A1: 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) ;
for A1, A2 being strict gate`2=den Boolean Circuit of n -BitGFA0Str (x,y) st ex f, g, h being ManySortedSet of NAT st
( n -BitGFA0Str (x,y) = f . n & A1 = g . n & 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 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) ) ) ) & ex f, g, h being ManySortedSet of NAT st
( n -BitGFA0Str (x,y) = f . n & A2 = g . n & 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 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) ) ) ) holds
A1 = A2 from CIRCCMB2:sch 21(A1);
hence for b<sub>1</sub>, b<sub>2</sub> being strict gate`2=den Boolean Circuit of n -BitGFA0Str (x,y) st ex f, g, h being ManySortedSet of NAT st
( n -BitGFA0Str (x,y) = f . n & b₁ = g . n & 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 ((x . (n + 1)),(y . (n + 1)),z)) & g . (n + 1) = A +* (BitGFA0Circ ((x . (n + 1)),(y . (n + 1)),z)) & h . (n + 1) = GFA0CarryOutput ((x . (n + 1)),(y . (n + 1)),z) ) ) ) & ex f, g, h being ManySortedSet of NAT st
( n -BitGFA0Str (x,y) = f . n & b₂ = g . n & 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 ((x . (n + 1)),(y . (n + 1)),z)) & g . (n + 1) = A +* (BitGFA0Circ ((x . (n + 1)),(y . (n + 1)),z)) & h . (n + 1) = GFA0CarryOutput ((x . (n + 1)),(y . (n + 1)),z) ) ) ) holds
b₁ = b₂ ; :: thesis: verum