set S0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE );
set A0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE );
set Sn = n -BitAdderStr x,y;
set o0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )];
deffunc H₁( set , Nat) -> Element of InnerVertices (MajorityStr (x . ($2 + 1)),(y . ($2 + 1)),$1) = MajorityOutput (x . ($2 + 1)),(y . ($2 + 1)),$1;
deffunc H₂( non empty ManySortedSign , set , Nat) -> ManySortedSign = $1 +* (BitAdderWithOverflowStr (x . ($3 + 1)),(y . ($3 + 1)),$2);
deffunc H₃( non empty ManySortedSign , non-empty MSAlgebra of $1, set , Nat) -> MSAlgebra of $1 +* (BitAdderWithOverflowStr (x . ($4 + 1)),(y . ($4 + 1)),$3) = $2 +* (BitAdderWithOverflowCirc (x . ($4 + 1)),(y . ($4 + 1)),$3);
A2: for S being non empty non void strict unsplit gate`1=arity gate`2isBoolean ManySortedSign
for z being set
for n being Nat holds
( H₂(S,z,n) is unsplit & H₂(S,z,n) is gate`1=arity & H<sub>2</sub>(S,z,n) is gate`2isBoolean & not H₂(S,z,n) is void & H₂(S,z,n) is strict ) ;
A3: 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 z being set
for n being Nat st S1 = H<sub>2</sub>(S,z,n) holds
H<sub>3</sub>(S,A,z,n) is strict gate`2=den Boolean Circuit of S1 ;
A4: for S being non empty ManySortedSign
for A being non-empty MSAlgebra of S
for z being set
for n being Nat holds H₃(S,A,z,n) is non-empty MSAlgebra of H₂(S,z,n) ;
A5: ex f, h being ManySortedSet of st
( n -BitAdderStr x,y = f . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] & ( for n being Nat
for S being non empty ManySortedSign
for z being set st S = f . n & z = h . n holds
( f . (n + 1) = H₂(S,z,n) & h . (n + 1) = H₁(z,n) ) ) ) by Def3;
thus ex A being strict gate`2=den Boolean Circuit of n -BitAdderStr x,y ex f, g, h being ManySortedSet of st
( n -BitAdderStr x,y = f . n & A = 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) = H₂(S,x,n) & g . (n + 1) = H₃(S,A,x,n) & h . (n + 1) = H₁(x,n) ) ) ) from CIRCCMB2:sch 19(A2, A5, A4, A3); :: thesis: verum