/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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rm: cannot remove ‘/tmp/muTerm386839851771476364.in.dec’: No such file or directory
rm: cannot remove ‘/tmp/muTerm386839851771476364.in.form’: No such file or directory
YES

Problem 1: 

(VAR M N X X1 X2 X3)
(RULES
U11(active(X1),X2,X3) -> U11(X1,X2,X3)
U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
U11(X1,active(X2),X3) -> U11(X1,X2,X3)
U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
U12(active(X1),X2,X3) -> U12(X1,X2,X3)
U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
U12(X1,active(X2),X3) -> U12(X1,X2,X3)
U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
U21(active(X1),X2,X3) -> U21(X1,X2,X3)
U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
U21(X1,active(X2),X3) -> U21(X1,X2,X3)
U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
U22(active(X1),X2,X3) -> U22(X1,X2,X3)
U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
U22(X1,active(X2),X3) -> U22(X1,X2,X3)
U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
active(U11(tt,M,N)) -> mark(U12(tt,M,N))
active(U12(tt,M,N)) -> mark(s(plus(N,M)))
active(U21(tt,M,N)) -> mark(U22(tt,M,N))
active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
active(plus(N,s(M))) -> mark(U11(tt,M,N))
active(plus(N,0)) -> mark(N)
active(x(N,s(M))) -> mark(U21(tt,M,N))
active(x(N,0)) -> mark(0)
mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
mark(s(X)) -> active(s(mark(X)))
mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
mark(0) -> active(0)
mark(tt) -> active(tt)
plus(active(X1),X2) -> plus(X1,X2)
plus(mark(X1),X2) -> plus(X1,X2)
plus(X1,active(X2)) -> plus(X1,X2)
plus(X1,mark(X2)) -> plus(X1,X2)
s(active(X)) -> s(X)
s(mark(X)) -> s(X)
x(active(X1),X2) -> x(X1,X2)
x(mark(X1),X2) -> x(X1,X2)
x(X1,active(X2)) -> x(X1,X2)
x(X1,mark(X2)) -> x(X1,X2)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(active(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
 U12#(active(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
 U21#(active(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
 U22#(active(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(mark(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
 ACTIVE(U11(tt,M,N)) -> U12#(tt,M,N)
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U12(tt,M,N)) -> PLUS(N,M)
 ACTIVE(U12(tt,M,N)) -> S(plus(N,M))
 ACTIVE(U21(tt,M,N)) -> U22#(tt,M,N)
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(U22(tt,M,N)) -> PLUS(x(N,M),N)
 ACTIVE(U22(tt,M,N)) -> X(N,M)
 ACTIVE(plus(N,s(M))) -> U11#(tt,M,N)
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> U21#(tt,M,N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 ACTIVE(x(N,0)) -> MARK(0)
 MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3)
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> U22#(mark(X1),X2,X3)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2))
 MARK(s(X)) -> ACTIVE(s(mark(X)))
 MARK(s(X)) -> MARK(X)
 MARK(s(X)) -> S(mark(X))
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
 MARK(x(X1,X2)) -> X(mark(X1),mark(X2))
 PLUS(active(X1),X2) -> PLUS(X1,X2)
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
 S(active(X)) -> S(X)
 S(mark(X)) -> S(X)
 X(active(X1),X2) -> X(X1,X2)
 X(mark(X1),X2) -> X(X1,X2)
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(active(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
 U12#(active(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
 U21#(active(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
 U22#(active(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(mark(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
 ACTIVE(U11(tt,M,N)) -> U12#(tt,M,N)
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U12(tt,M,N)) -> PLUS(N,M)
 ACTIVE(U12(tt,M,N)) -> S(plus(N,M))
 ACTIVE(U21(tt,M,N)) -> U22#(tt,M,N)
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(U22(tt,M,N)) -> PLUS(x(N,M),N)
 ACTIVE(U22(tt,M,N)) -> X(N,M)
 ACTIVE(plus(N,s(M))) -> U11#(tt,M,N)
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> U21#(tt,M,N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 ACTIVE(x(N,0)) -> MARK(0)
 MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3)
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> U21#(mark(X1),X2,X3)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> U22#(mark(X1),X2,X3)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2))
 MARK(s(X)) -> ACTIVE(s(mark(X)))
 MARK(s(X)) -> MARK(X)
 MARK(s(X)) -> S(mark(X))
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
 MARK(x(X1,X2)) -> X(mark(X1),mark(X2))
 PLUS(active(X1),X2) -> PLUS(X1,X2)
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
 S(active(X)) -> S(X)
 S(mark(X)) -> S(X)
 X(active(X1),X2) -> X(X1,X2)
 X(mark(X1),X2) -> X(X1,X2)
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 X(active(X1),X2) -> X(X1,X2)
 X(mark(X1),X2) -> X(X1,X2)
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 S(active(X)) -> S(X)
 S(mark(X)) -> S(X)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 PLUS(active(X1),X2) -> PLUS(X1,X2)
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 U22#(active(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(mark(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 U21#(active(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 U12#(active(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 U11#(active(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> ACTIVE(s(mark(X)))
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)


The problem is decomposed in 8 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 X(active(X1),X2) -> X(X1,X2)
 X(mark(X1),X2) -> X(X1,X2)
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(X) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.1: 

Subterm Processor:
-> Pairs:
 X(X1,active(X2)) -> X(X1,X2)
 X(X1,mark(X2)) -> X(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(X) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 S(active(X)) -> S(X)
 S(mark(X)) -> S(X)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(S) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 PLUS(active(X1),X2) -> PLUS(X1,X2)
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(PLUS) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.3: 

Subterm Processor:
-> Pairs:
 PLUS(X1,active(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(PLUS) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 U22#(active(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(mark(X1),X2,X3) -> U22#(X1,X2,X3)
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U22#) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.4: 

Subterm Processor:
-> Pairs:
 U22#(X1,active(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,mark(X2),X3) -> U22#(X1,X2,X3)
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U22#) = 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.4: 

Subterm Processor:
-> Pairs:
 U22#(X1,X2,active(X3)) -> U22#(X1,X2,X3)
 U22#(X1,X2,mark(X3)) -> U22#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U22#) = 3

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 U21#(active(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(mark(X1),X2,X3) -> U21#(X1,X2,X3)
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U21#) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.5: 

Subterm Processor:
-> Pairs:
 U21#(X1,active(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,mark(X2),X3) -> U21#(X1,X2,X3)
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U21#) = 2

Problem 1.5: 

SCC Processor:
-> Pairs:
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.5: 

Subterm Processor:
-> Pairs:
 U21#(X1,X2,active(X3)) -> U21#(X1,X2,X3)
 U21#(X1,X2,mark(X3)) -> U21#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U21#) = 3

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 U12#(active(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U12#) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.6: 

Subterm Processor:
-> Pairs:
 U12#(X1,active(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3)
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U12#) = 2

Problem 1.6: 

SCC Processor:
-> Pairs:
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.6: 

Subterm Processor:
-> Pairs:
 U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3)
 U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U12#) = 3

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Subterm Processor:
-> Pairs:
 U11#(active(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U11#) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.7: 

Subterm Processor:
-> Pairs:
 U11#(X1,active(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3)
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U11#) = 2

Problem 1.7: 

SCC Processor:
-> Pairs:
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.7: 

Subterm Processor:
-> Pairs:
 U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3)
 U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(U11#) = 3

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> ACTIVE(s(mark(X)))
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 1
[U12](X1,X2,X3) = 1
[U21](X1,X2,X3) = 1
[U22](X1,X2,X3) = 1
[active](X) = 2.X
[mark](X) = 2
[plus](X1,X2) = 1
[s](X) = 0
[x](X1,X2) = 1
[0] = 1
[tt] = 1
[ACTIVE](X) = X + 1
[MARK](X) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N))
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2
[U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + 2.X2 + 2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
[0] = 0
[tt] = 2
[ACTIVE](X) = 2.X.X + 2.X + 2
[MARK](X) = 2.X.X + 2.X + 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 0
[U12](X1,X2,X3) = 1
[U21](X1,X2,X3) = 1
[U22](X1,X2,X3) = 1
[active](X) = 1
[mark](X) = 1
[plus](X1,X2) = 1
[s](X) = 0
[x](X1,X2) = 1
[0] = 2
[tt] = 2
[ACTIVE](X) = 2.X
[MARK](X) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M)))
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2
[U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
[U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + 2.X2 + 2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
[0] = 2
[tt] = 2
[ACTIVE](X) = 2.X.X + 2.X + 2
[MARK](X) = 2.X.X + 2.X + 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3))
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 1
[U12](X1,X2,X3) = 0
[U21](X1,X2,X3) = 1
[U22](X1,X2,X3) = 1
[active](X) = 2
[mark](X) = 2
[plus](X1,X2) = 1
[s](X) = 2
[x](X1,X2) = 1
[0] = 0
[tt] = 0
[ACTIVE](X) = X + 1
[MARK](X) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U21(tt,M,N)) -> MARK(U22(tt,M,N))
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2
[U12](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1 + X2 + X3 + 2
[U21](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + X2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1
[0] = 2
[tt] = 0
[ACTIVE](X) = 2.X + 2
[MARK](X) = 2.X + 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> ACTIVE(U21(mark(X1),X2,X3))
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = 1
[U21](X1,X2,X3) = 1
[U22](X1,X2,X3) = 2
[active](X) = 2
[mark](X) = 2
[plus](X1,X2) = 2
[s](X) = 1
[x](X1,X2) = 2
[0] = 1
[tt] = 2
[ACTIVE](X) = X
[MARK](X) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(U22(tt,M,N)) -> MARK(plus(x(N,M),N))
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
[U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
[U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
[U22](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + X2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1
[0] = 2
[tt] = 0
[ACTIVE](X) = 2.X + 1
[MARK](X) = 2.X + 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> ACTIVE(U22(mark(X1),X2,X3))
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = 0
[U21](X1,X2,X3) = 2
[U22](X1,X2,X3) = 1
[active](X) = 2
[mark](X) = 2
[plus](X1,X2) = 2
[s](X) = 2
[x](X1,X2) = 2
[0] = 0
[tt] = 2
[ACTIVE](X) = X
[MARK](X) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N))
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 1
[U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[U22](X1,X2,X3) = X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + 2.X2 + 2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
[0] = 2
[tt] = 2
[ACTIVE](X) = 2.X.X + 2.X + 2
[MARK](X) = 2.X.X + 2.X + 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(plus(N,0)) -> MARK(N)
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
[U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
[U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X3 + 2
[U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + X3 + 1
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + 2.X2 + 2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
[0] = 2
[tt] = 2
[ACTIVE](X) = X + 2
[MARK](X) = X + 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2)))
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = 0
[U21](X1,X2,X3) = 2
[U22](X1,X2,X3) = 2
[active](X) = 2
[mark](X) = 2
[plus](X1,X2) = 1
[s](X) = 0
[x](X1,X2) = 2
[0] = 2
[tt] = 0
[ACTIVE](X) = X
[MARK](X) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(x(N,s(M))) -> MARK(U21(tt,M,N))
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
-> Usable rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + X2 + X3
[U12](X1,X2,X3) = X1 + X2 + X3 + 2
[U21](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1
[U22](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1
[active](X) = X
[mark](X) = X
[plus](X1,X2) = X1 + X2 + 2
[s](X) = X + 2
[x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2
[0] = 1
[tt] = 2
[ACTIVE](X) = 2.X.X + 2.X + 2
[MARK](X) = 2.X.X + 2.X + 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> ACTIVE(x(mark(X1),mark(X2)))
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
->->-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)

Problem 1.8: 

Subterm Processor:
-> Pairs:
 MARK(U11(X1,X2,X3)) -> MARK(X1)
 MARK(U12(X1,X2,X3)) -> MARK(X1)
 MARK(U21(X1,X2,X3)) -> MARK(X1)
 MARK(U22(X1,X2,X3)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X1)
 MARK(plus(X1,X2)) -> MARK(X2)
 MARK(s(X)) -> MARK(X)
 MARK(x(X1,X2)) -> MARK(X1)
 MARK(x(X1,X2)) -> MARK(X2)
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Projection:
 pi(MARK) = 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U22(active(X1),X2,X3) -> U22(X1,X2,X3)
 U22(mark(X1),X2,X3) -> U22(X1,X2,X3)
 U22(X1,active(X2),X3) -> U22(X1,X2,X3)
 U22(X1,mark(X2),X3) -> U22(X1,X2,X3)
 U22(X1,X2,active(X3)) -> U22(X1,X2,X3)
 U22(X1,X2,mark(X3)) -> U22(X1,X2,X3)
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U21(tt,M,N)) -> mark(U22(tt,M,N))
 active(U22(tt,M,N)) -> mark(plus(x(N,M),N))
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(x(N,s(M))) -> mark(U21(tt,M,N))
 active(x(N,0)) -> mark(0)
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2,X3)) -> active(U22(mark(X1),X2,X3))
 mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2)))
 mark(s(X)) -> active(s(mark(X)))
 mark(x(X1,X2)) -> active(x(mark(X1),mark(X2)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1),X2) -> plus(X1,X2)
 plus(mark(X1),X2) -> plus(X1,X2)
 plus(X1,active(X2)) -> plus(X1,X2)
 plus(X1,mark(X2)) -> plus(X1,X2)
 s(active(X)) -> s(X)
 s(mark(X)) -> s(X)
 x(active(X1),X2) -> x(X1,X2)
 x(mark(X1),X2) -> x(X1,X2)
 x(X1,active(X2)) -> x(X1,X2)
 x(X1,mark(X2)) -> x(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.