Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(2nd1(cons2(X, XS))) -> HEAD1(XS)
TAKE2(X1, mark1(X2)) -> TAKE2(X1, X2)
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(head1(X)) -> PROPER1(X)
S1(mark1(X)) -> S1(X)
PROPER1(take2(X1, X2)) -> PROPER1(X2)
ACTIVE1(take2(s1(N), cons2(X, XS))) -> TAKE2(N, XS)
ACTIVE1(2nd1(X)) -> ACTIVE1(X)
TOP1(mark1(X)) -> TOP1(proper1(X))
2ND1(mark1(X)) -> 2ND1(X)
HEAD1(mark1(X)) -> HEAD1(X)
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)
2ND1(ok1(X)) -> 2ND1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
PROPER1(take2(X1, X2)) -> PROPER1(X1)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> TAKE2(active1(X1), X2)
PROPER1(2nd1(X)) -> 2ND1(proper1(X))
HEAD1(ok1(X)) -> HEAD1(X)
PROPER1(take2(X1, X2)) -> TAKE2(proper1(X1), proper1(X2))
TOP1(mark1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
PROPER1(2nd1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(2nd1(X)) -> 2ND1(active1(X))
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(s1(N), cons2(X, XS))) -> SEL2(N, XS)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(head1(X)) -> HEAD1(active1(X))
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
TAKE2(mark1(X1), X2) -> TAKE2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
PROPER1(head1(X)) -> HEAD1(proper1(X))
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(from1(X)) -> S1(X)
ACTIVE1(take2(X1, X2)) -> TAKE2(X1, active1(X2))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(take2(s1(N), cons2(X, XS))) -> CONS2(X, take2(N, XS))
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(2nd1(cons2(X, XS))) -> HEAD1(XS)
TAKE2(X1, mark1(X2)) -> TAKE2(X1, X2)
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(head1(X)) -> PROPER1(X)
S1(mark1(X)) -> S1(X)
PROPER1(take2(X1, X2)) -> PROPER1(X2)
ACTIVE1(take2(s1(N), cons2(X, XS))) -> TAKE2(N, XS)
ACTIVE1(2nd1(X)) -> ACTIVE1(X)
TOP1(mark1(X)) -> TOP1(proper1(X))
2ND1(mark1(X)) -> 2ND1(X)
HEAD1(mark1(X)) -> HEAD1(X)
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)
2ND1(ok1(X)) -> 2ND1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
PROPER1(take2(X1, X2)) -> PROPER1(X1)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> TAKE2(active1(X1), X2)
PROPER1(2nd1(X)) -> 2ND1(proper1(X))
HEAD1(ok1(X)) -> HEAD1(X)
PROPER1(take2(X1, X2)) -> TAKE2(proper1(X1), proper1(X2))
TOP1(mark1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
PROPER1(2nd1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(2nd1(X)) -> 2ND1(active1(X))
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(s1(N), cons2(X, XS))) -> SEL2(N, XS)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(head1(X)) -> HEAD1(active1(X))
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
TAKE2(mark1(X1), X2) -> TAKE2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
PROPER1(head1(X)) -> HEAD1(proper1(X))
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(from1(X)) -> S1(X)
ACTIVE1(take2(X1, X2)) -> TAKE2(X1, active1(X2))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(take2(s1(N), cons2(X, XS))) -> CONS2(X, take2(N, XS))
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 10 SCCs with 25 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.

SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(SEL2(x1, x2)) = x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.

SEL2(mark1(X1), X2) -> SEL2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(SEL2(x1, x2)) = x1   
POL(mark1(x1)) = 3·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL2(mark1(X1), X2) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SEL2(x1, x2)) = 2·x1   
POL(mark1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TAKE2(mark1(X1), X2) -> TAKE2(X1, X2)
TAKE2(X1, mark1(X2)) -> TAKE2(X1, X2)
TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TAKE2(mark1(X1), X2) -> TAKE2(X1, X2)
The remaining pairs can at least be oriented weakly.

TAKE2(X1, mark1(X2)) -> TAKE2(X1, X2)
TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(TAKE2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TAKE2(X1, mark1(X2)) -> TAKE2(X1, X2)
TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TAKE2(X1, mark1(X2)) -> TAKE2(X1, X2)
The remaining pairs can at least be oriented weakly.

TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(TAKE2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TAKE2(ok1(X1), ok1(X2)) -> TAKE2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(TAKE2(x1, x2)) = 2·x1 + 2·x2   
POL(ok1(x1)) = 2 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

2ND1(mark1(X)) -> 2ND1(X)
2ND1(ok1(X)) -> 2ND1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2ND1(mark1(X)) -> 2ND1(X)
The remaining pairs can at least be oriented weakly.

2ND1(ok1(X)) -> 2ND1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ND1(x1)) = x1   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

2ND1(ok1(X)) -> 2ND1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2ND1(ok1(X)) -> 2ND1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(2ND1(x1)) = 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

HEAD1(ok1(X)) -> HEAD1(X)
HEAD1(mark1(X)) -> HEAD1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


HEAD1(ok1(X)) -> HEAD1(X)
The remaining pairs can at least be oriented weakly.

HEAD1(mark1(X)) -> HEAD1(X)
Used ordering: Polynomial interpretation [21]:

POL(HEAD1(x1)) = x1   
POL(mark1(x1)) = 3·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

HEAD1(mark1(X)) -> HEAD1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


HEAD1(mark1(X)) -> HEAD1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(HEAD1(x1)) = 2·x1   
POL(mark1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(ok1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.

S1(mark1(X)) -> S1(X)
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = x1   
POL(mark1(x1)) = 3·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = 2·x1   
POL(mark1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = x1 + 2·x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = 2·x1 + 2·x2   
POL(ok1(x1)) = 2 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FROM1(mark1(X)) -> FROM1(X)
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(mark1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.

FROM1(ok1(X)) -> FROM1(X)
Used ordering: Polynomial interpretation [21]:

POL(FROM1(x1)) = x1   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(ok1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FROM1(x1)) = 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(take2(X1, X2)) -> PROPER1(X2)
PROPER1(take2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(2nd1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(take2(X1, X2)) -> PROPER1(X2)
PROPER1(take2(X1, X2)) -> PROPER1(X1)
The remaining pairs can at least be oriented weakly.

PROPER1(s1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(2nd1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = 3·x1 + 3·x2   
POL(from1(x1)) = 3·x1   
POL(head1(x1)) = 3·x1   
POL(s1(x1)) = 3·x1   
POL(sel2(x1, x2)) = 3·x1 + 3·x2   
POL(take2(x1, x2)) = 1 + 3·x1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(s1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = 3·x1 + 3·x2   
POL(from1(x1)) = 3·x1   
POL(head1(x1)) = 3·x1   
POL(s1(x1)) = 1 + 2·x1   
POL(sel2(x1, x2)) = 3·x1 + 3·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(2nd1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = 3·x1 + 3·x2   
POL(from1(x1)) = 3·x1   
POL(head1(x1)) = 3·x1   
POL(sel2(x1, x2)) = 1 + 2·x1 + 3·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(from1(X)) -> PROPER1(X)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = 1 + 2·x1 + 3·x2   
POL(from1(x1)) = 3·x1   
POL(head1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(from1(X)) -> PROPER1(X)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(from1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(PROPER1(x1)) = x1   
POL(from1(x1)) = 1 + 2·x1   
POL(head1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(head1(X)) -> PROPER1(X)
PROPER1(2nd1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(head1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(2nd1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(PROPER1(x1)) = x1   
POL(head1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER1(2nd1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(2nd1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 1 + 2·x1   
POL(PROPER1(x1)) = 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2nd1(X)) -> ACTIVE1(X)
ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(2nd1(X)) -> ACTIVE1(X)
ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = 3·x1   
POL(from1(x1)) = 3·x1   
POL(head1(x1)) = 3·x1   
POL(s1(x1)) = 3·x1   
POL(sel2(x1, x2)) = 1 + 3·x1 + 2·x2   
POL(take2(x1, x2)) = 3·x1 + 3·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(2nd1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(head1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(2nd1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 3·x1   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = 3·x1   
POL(from1(x1)) = 3·x1   
POL(head1(x1)) = 1 + 2·x1   
POL(s1(x1)) = 3·x1   
POL(take2(x1, x2)) = 3·x1 + 3·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(2nd1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(2nd1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 1 + 2·x1   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = 3·x1   
POL(from1(x1)) = 3·x1   
POL(s1(x1)) = 3·x1   
POL(take2(x1, x2)) = 3·x1 + 3·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = 1 + 2·x1   
POL(from1(x1)) = 3·x1   
POL(s1(x1)) = 3·x1   
POL(take2(x1, x2)) = 3·x1 + 3·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(take2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(take2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = x1   
POL(from1(x1)) = 3·x1   
POL(s1(x1)) = 3·x1   
POL(take2(x1, x2)) = 1 + 3·x1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(from1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = x1   
POL(from1(x1)) = 1 + 2·x1   
POL(s1(x1)) = 3·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(s1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = 2·x1   
POL(s1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ PisEmptyProof
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, XS))) -> mark1(X)
active1(2nd1(cons2(X, XS))) -> mark1(head1(XS))
active1(take2(0, XS)) -> mark1(nil)
active1(take2(s1(N), cons2(X, XS))) -> mark1(cons2(X, take2(N, XS)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(head1(X)) -> head1(active1(X))
active1(2nd1(X)) -> 2nd1(active1(X))
active1(take2(X1, X2)) -> take2(active1(X1), X2)
active1(take2(X1, X2)) -> take2(X1, active1(X2))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
head1(mark1(X)) -> mark1(head1(X))
2nd1(mark1(X)) -> mark1(2nd1(X))
take2(mark1(X1), X2) -> mark1(take2(X1, X2))
take2(X1, mark1(X2)) -> mark1(take2(X1, X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(head1(X)) -> head1(proper1(X))
proper1(2nd1(X)) -> 2nd1(proper1(X))
proper1(take2(X1, X2)) -> take2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(nil) -> ok1(nil)
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
head1(ok1(X)) -> ok1(head1(X))
2nd1(ok1(X)) -> ok1(2nd1(X))
take2(ok1(X1), ok1(X2)) -> ok1(take2(X1, X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.