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 of $1, set , Nat) -> MSAlgebra of $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 of S
for x being set
for n being Nat holds H₂(S,A,x,n) is non-empty MSAlgebra of 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 of 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 of 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 of 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 of 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