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


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YES

Problem 1: 

(VAR X X1 X2 Y Z)
(RULES
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
active(2nd(X)) -> 2nd(active(X))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(cons1(X1,X2)) -> cons1(active(X1),X2)
active(cons1(X1,X2)) -> cons1(X1,active(X2))
active(from(X)) -> from(active(X))
active(from(X)) -> mark(cons(X,from(s(X))))
active(s(X)) -> s(active(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
cons1(mark(X1),X2) -> mark(cons1(X1,X2))
cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 2ND(mark(X)) -> 2ND(X)
 2ND(ok(X)) -> 2ND(X)
 ACTIVE(2nd(cons(X,X1))) -> 2ND(cons1(X,X1))
 ACTIVE(2nd(cons(X,X1))) -> CONS1(X,X1)
 ACTIVE(2nd(X)) -> 2ND(active(X))
 ACTIVE(2nd(X)) -> ACTIVE(X)
 ACTIVE(cons(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons(X1,X2)) -> CONS(active(X1),X2)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X2)
 ACTIVE(cons1(X1,X2)) -> CONS1(active(X1),X2)
 ACTIVE(cons1(X1,X2)) -> CONS1(X1,active(X2))
 ACTIVE(from(X)) -> ACTIVE(X)
 ACTIVE(from(X)) -> CONS(X,from(s(X)))
 ACTIVE(from(X)) -> FROM(active(X))
 ACTIVE(from(X)) -> FROM(s(X))
 ACTIVE(from(X)) -> S(X)
 ACTIVE(s(X)) -> ACTIVE(X)
 ACTIVE(s(X)) -> S(active(X))
 CONS(mark(X1),X2) -> CONS(X1,X2)
 CONS(ok(X1),ok(X2)) -> CONS(X1,X2)
 CONS1(mark(X1),X2) -> CONS1(X1,X2)
 CONS1(ok(X1),ok(X2)) -> CONS1(X1,X2)
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
 FROM(mark(X)) -> FROM(X)
 FROM(ok(X)) -> FROM(X)
 PROPER(2nd(X)) -> 2ND(proper(X))
 PROPER(2nd(X)) -> PROPER(X)
 PROPER(cons(X1,X2)) -> CONS(proper(X1),proper(X2))
 PROPER(cons(X1,X2)) -> PROPER(X1)
 PROPER(cons(X1,X2)) -> PROPER(X2)
 PROPER(cons1(X1,X2)) -> CONS1(proper(X1),proper(X2))
 PROPER(cons1(X1,X2)) -> PROPER(X1)
 PROPER(cons1(X1,X2)) -> PROPER(X2)
 PROPER(from(X)) -> FROM(proper(X))
 PROPER(from(X)) -> PROPER(X)
 PROPER(s(X)) -> PROPER(X)
 PROPER(s(X)) -> S(proper(X))
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
 TOP(mark(X)) -> PROPER(X)
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> ACTIVE(X)
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Problem 1: 

SCC Processor:
-> Pairs:
 2ND(mark(X)) -> 2ND(X)
 2ND(ok(X)) -> 2ND(X)
 ACTIVE(2nd(cons(X,X1))) -> 2ND(cons1(X,X1))
 ACTIVE(2nd(cons(X,X1))) -> CONS1(X,X1)
 ACTIVE(2nd(X)) -> 2ND(active(X))
 ACTIVE(2nd(X)) -> ACTIVE(X)
 ACTIVE(cons(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons(X1,X2)) -> CONS(active(X1),X2)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X2)
 ACTIVE(cons1(X1,X2)) -> CONS1(active(X1),X2)
 ACTIVE(cons1(X1,X2)) -> CONS1(X1,active(X2))
 ACTIVE(from(X)) -> ACTIVE(X)
 ACTIVE(from(X)) -> CONS(X,from(s(X)))
 ACTIVE(from(X)) -> FROM(active(X))
 ACTIVE(from(X)) -> FROM(s(X))
 ACTIVE(from(X)) -> S(X)
 ACTIVE(s(X)) -> ACTIVE(X)
 ACTIVE(s(X)) -> S(active(X))
 CONS(mark(X1),X2) -> CONS(X1,X2)
 CONS(ok(X1),ok(X2)) -> CONS(X1,X2)
 CONS1(mark(X1),X2) -> CONS1(X1,X2)
 CONS1(ok(X1),ok(X2)) -> CONS1(X1,X2)
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
 FROM(mark(X)) -> FROM(X)
 FROM(ok(X)) -> FROM(X)
 PROPER(2nd(X)) -> 2ND(proper(X))
 PROPER(2nd(X)) -> PROPER(X)
 PROPER(cons(X1,X2)) -> CONS(proper(X1),proper(X2))
 PROPER(cons(X1,X2)) -> PROPER(X1)
 PROPER(cons(X1,X2)) -> PROPER(X2)
 PROPER(cons1(X1,X2)) -> CONS1(proper(X1),proper(X2))
 PROPER(cons1(X1,X2)) -> PROPER(X1)
 PROPER(cons1(X1,X2)) -> PROPER(X2)
 PROPER(from(X)) -> FROM(proper(X))
 PROPER(from(X)) -> PROPER(X)
 PROPER(s(X)) -> PROPER(X)
 PROPER(s(X)) -> S(proper(X))
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
 TOP(mark(X)) -> PROPER(X)
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> ACTIVE(X)
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 FROM(mark(X)) -> FROM(X)
 FROM(ok(X)) -> FROM(X)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 CONS1(mark(X1),X2) -> CONS1(X1,X2)
 CONS1(ok(X1),ok(X2)) -> CONS1(X1,X2)
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 CONS(mark(X1),X2) -> CONS(X1,X2)
 CONS(ok(X1),ok(X2)) -> CONS(X1,X2)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 2ND(mark(X)) -> 2ND(X)
 2ND(ok(X)) -> 2ND(X)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 ACTIVE(2nd(X)) -> ACTIVE(X)
 ACTIVE(cons(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X2)
 ACTIVE(from(X)) -> ACTIVE(X)
 ACTIVE(s(X)) -> ACTIVE(X)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 PROPER(2nd(X)) -> PROPER(X)
 PROPER(cons(X1,X2)) -> PROPER(X1)
 PROPER(cons(X1,X2)) -> PROPER(X2)
 PROPER(cons1(X1,X2)) -> PROPER(X1)
 PROPER(cons1(X1,X2)) -> PROPER(X2)
 PROPER(from(X)) -> PROPER(X)
 PROPER(s(X)) -> PROPER(X)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> TOP(active(X))
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))


The problem is decomposed in 8 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(S) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 FROM(mark(X)) -> FROM(X)
 FROM(ok(X)) -> FROM(X)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(FROM) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 CONS1(mark(X1),X2) -> CONS1(X1,X2)
 CONS1(ok(X1),ok(X2)) -> CONS1(X1,X2)
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(CONS1) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Problem 1.3: 

Subterm Processor:
-> Pairs:
 CONS1(X1,mark(X2)) -> CONS1(X1,X2)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(CONS1) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 CONS(mark(X1),X2) -> CONS(X1,X2)
 CONS(ok(X1),ok(X2)) -> CONS(X1,X2)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(CONS) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 2ND(mark(X)) -> 2ND(X)
 2ND(ok(X)) -> 2ND(X)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(2ND) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 ACTIVE(2nd(X)) -> ACTIVE(X)
 ACTIVE(cons(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X1)
 ACTIVE(cons1(X1,X2)) -> ACTIVE(X2)
 ACTIVE(from(X)) -> ACTIVE(X)
 ACTIVE(s(X)) -> ACTIVE(X)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(ACTIVE) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Subterm Processor:
-> Pairs:
 PROPER(2nd(X)) -> PROPER(X)
 PROPER(cons(X1,X2)) -> PROPER(X1)
 PROPER(cons(X1,X2)) -> PROPER(X2)
 PROPER(cons1(X1,X2)) -> PROPER(X1)
 PROPER(cons1(X1,X2)) -> PROPER(X2)
 PROPER(from(X)) -> PROPER(X)
 PROPER(s(X)) -> PROPER(X)
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(PROPER) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
-> Usable rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[2nd](X) = 2.X + 2
[active](X) = 2.X + 2
[cons](X1,X2) = 2.X1 + X2 + 2
[cons1](X1,X2) = 2.X1 + 2.X2 + 2
[from](X) = 2.X + 2
[proper](X) = X
[s](X) = X
[mark](X) = X + 2
[ok](X) = 2.X + 2
[TOP](X) = X

Problem 1.8: 

SCC Processor:
-> Pairs:
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TOP(ok(X)) -> TOP(active(X))
->->-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Problem 1.8: 

Reduction Pair Processor:
-> Pairs:
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
-> Usable rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[2nd](X) = 2.X
[active](X) = 2.X
[cons](X1,X2) = 2.X1 + 2
[cons1](X1,X2) = 2.X1 + 2.X2
[from](X) = 2.X + 2
[s](X) = 2.X + 2
[mark](X) = 2
[ok](X) = 2.X + 2
[TOP](X) = 2.X

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(mark(X)) -> mark(2nd(X))
 2nd(ok(X)) -> ok(2nd(X))
 active(2nd(cons(X,X1))) -> mark(2nd(cons1(X,X1)))
 active(2nd(cons1(X,cons(Y,Z)))) -> mark(Y)
 active(2nd(X)) -> 2nd(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(active(X1),X2)
 active(cons1(X1,X2)) -> cons1(X1,active(X2))
 active(from(X)) -> from(active(X))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(s(X)) -> s(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 cons1(mark(X1),X2) -> mark(cons1(X1,X2))
 cons1(ok(X1),ok(X2)) -> ok(cons1(X1,X2))
 cons1(X1,mark(X2)) -> mark(cons1(X1,X2))
 from(mark(X)) -> mark(from(X))
 from(ok(X)) -> ok(from(X))
 proper(2nd(X)) -> 2nd(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(cons1(X1,X2)) -> cons1(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 proper(s(X)) -> s(proper(X))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.