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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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(g1(X)) -> ACTIVE1(X)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(f1(X)) -> ACTIVE1(X)
ACTIVE1(f1(X)) -> G1(X)
ACTIVE1(g1(s1(X))) -> G1(X)
ACTIVE1(g1(s1(X))) -> S1(g1(X))
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
S1(mark1(X)) -> S1(X)
G1(mark1(X)) -> G1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
PROPER1(f1(X)) -> PROPER1(X)
ACTIVE1(sel2(s1(X), cons2(Y, Z))) -> SEL2(X, Z)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
PROPER1(g1(X)) -> G1(proper1(X))
PROPER1(g1(X)) -> PROPER1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(f1(X)) -> F1(active1(X))
ACTIVE1(g1(s1(X))) -> S1(s1(g1(X)))
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
F1(mark1(X)) -> F1(X)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(g1(X)) -> G1(active1(X))
ACTIVE1(f1(X)) -> F1(g1(X))
TOP1(ok1(X)) -> TOP1(active1(X))
F1(ok1(X)) -> F1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
ACTIVE1(g1(0)) -> S1(0)
G1(ok1(X)) -> G1(X)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(f1(X)) -> CONS2(X, f1(g1(X)))
PROPER1(f1(X)) -> F1(proper1(X))

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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(g1(X)) -> ACTIVE1(X)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(f1(X)) -> ACTIVE1(X)
ACTIVE1(f1(X)) -> G1(X)
ACTIVE1(g1(s1(X))) -> G1(X)
ACTIVE1(g1(s1(X))) -> S1(g1(X))
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
S1(mark1(X)) -> S1(X)
G1(mark1(X)) -> G1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
PROPER1(f1(X)) -> PROPER1(X)
ACTIVE1(sel2(s1(X), cons2(Y, Z))) -> SEL2(X, Z)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
PROPER1(g1(X)) -> G1(proper1(X))
PROPER1(g1(X)) -> PROPER1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(f1(X)) -> F1(active1(X))
ACTIVE1(g1(s1(X))) -> S1(s1(g1(X)))
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
F1(mark1(X)) -> F1(X)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(g1(X)) -> G1(active1(X))
ACTIVE1(f1(X)) -> F1(g1(X))
TOP1(ok1(X)) -> TOP1(active1(X))
F1(ok1(X)) -> F1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
ACTIVE1(g1(0)) -> S1(0)
G1(ok1(X)) -> G1(X)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(f1(X)) -> CONS2(X, f1(g1(X)))
PROPER1(f1(X)) -> F1(proper1(X))

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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 8 SCCs with 21 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ 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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.

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

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

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

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

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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.
none
Used ordering: Polynomial interpretation [21]:

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

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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)
S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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

G1(mark1(X)) -> G1(X)
G1(ok1(X)) -> G1(X)

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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.


G1(mark1(X)) -> G1(X)
G1(ok1(X)) -> G1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(G1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



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

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

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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.

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = x2   
POL(mark1(x1)) = 0   
POL(ok1(x1)) = 1 + 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

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

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

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + 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

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

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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

F1(mark1(X)) -> F1(X)
F1(ok1(X)) -> F1(X)

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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.


F1(mark1(X)) -> F1(X)
F1(ok1(X)) -> F1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(F1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



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

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

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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

PROPER1(g1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(f1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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(g1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(f1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = 1 + x1 + x2   
POL(f1(x1)) = x1   
POL(g1(x1)) = 1 + x1   
POL(s1(x1)) = 1 + x1   
POL(sel2(x1, x2)) = 1 + x1 + x2   

The following usable rules [14] were oriented: none



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

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

PROPER1(f1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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(f1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = x1   
POL(f1(x1)) = 1 + 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

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

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(g1(X)) -> ACTIVE1(X)
ACTIVE1(f1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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(g1(X)) -> ACTIVE1(X)
ACTIVE1(f1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = 1 + x1   
POL(f1(x1)) = 1 + x1   
POL(g1(x1)) = 1 + x1   
POL(s1(x1)) = 1 + x1   
POL(sel2(x1, x2)) = 1 + x1 + x2   

The following usable rules [14] were oriented: none



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

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

active1(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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

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(f1(X)) -> mark1(cons2(X, f1(g1(X))))
active1(g1(0)) -> mark1(s1(0))
active1(g1(s1(X))) -> mark1(s1(s1(g1(X))))
active1(sel2(0, cons2(X, Y))) -> mark1(X)
active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(f1(X)) -> f1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(g1(X)) -> g1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
f1(mark1(X)) -> mark1(f1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
g1(mark1(X)) -> mark1(g1(X))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
proper1(f1(X)) -> f1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(g1(X)) -> g1(proper1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
f1(ok1(X)) -> ok1(f1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
g1(ok1(X)) -> ok1(g1(X))
s1(ok1(X)) -> ok1(s1(X))
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.