set S0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set A0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE));
set Sn = n -BitSubtracterStr (x,y);
set o0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)];
deffunc H1( set , Nat) -> Element of InnerVertices (BorrowStr ((x . ($2 + 1)),(y . ($2 + 1)),$1)) = BorrowOutput ((x . ($2 + 1)),(y . ($2 + 1)),$1);
deffunc H2( non empty ManySortedSign , set , Nat) -> ManySortedSign = $1 +* (BitSubtracterWithBorrowStr ((x . ($3 + 1)),(y . ($3 + 1)),$2));
deffunc H3( non empty ManySortedSign , non-empty MSAlgebra over $1, set , Nat) -> MSAlgebra over $1 +* (BitSubtracterWithBorrowStr ((x . ($4 + 1)),(y . ($4 + 1)),$3)) = $2 +* (BitSubtracterWithBorrowCirc ((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 & H2(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 = H2(S,z,n) holds
H3(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 over S
for z being set
for n being Nat holds H3(S,A,z,n) is non-empty MSAlgebra over H2(S,z,n)
;
A5:
ex f, h being ManySortedSet of NAT st
( n -BitSubtracterStr (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 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 Def1;
thus
ex A being strict gate`2=den Boolean Circuit of n -BitSubtracterStr (x,y) ex f, g, h being ManySortedSet of NAT st
( n -BitSubtracterStr (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) = 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); verum