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(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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(f3(x, y, z)) -> ACTIVE1(z)
PROPER1(f3(x, y, z)) -> PROPER1(x)
TOP1(mark1(x)) -> PROPER1(x)
PROPER1(f3(x, y, z)) -> PROPER1(z)
TOP1(ok1(x)) -> ACTIVE1(x)
ACTIVE1(f3(x, y, z)) -> F3(x, y, active1(z))
ACTIVE1(f3(b, c, x)) -> F3(x, x, x)
TOP1(ok1(x)) -> TOP1(active1(x))
TOP1(mark1(x)) -> TOP1(proper1(x))
PROPER1(f3(x, y, z)) -> PROPER1(y)
PROPER1(f3(x, y, z)) -> F3(proper1(x), proper1(y), proper1(z))
F3(x, y, mark1(z)) -> F3(x, y, z)
F3(ok1(x), ok1(y), ok1(z)) -> F3(x, y, z)

The TRS R consists of the following rules:

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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(f3(x, y, z)) -> ACTIVE1(z)
PROPER1(f3(x, y, z)) -> PROPER1(x)
TOP1(mark1(x)) -> PROPER1(x)
PROPER1(f3(x, y, z)) -> PROPER1(z)
TOP1(ok1(x)) -> ACTIVE1(x)
ACTIVE1(f3(x, y, z)) -> F3(x, y, active1(z))
ACTIVE1(f3(b, c, x)) -> F3(x, x, x)
TOP1(ok1(x)) -> TOP1(active1(x))
TOP1(mark1(x)) -> TOP1(proper1(x))
PROPER1(f3(x, y, z)) -> PROPER1(y)
PROPER1(f3(x, y, z)) -> F3(proper1(x), proper1(y), proper1(z))
F3(x, y, mark1(z)) -> F3(x, y, z)
F3(ok1(x), ok1(y), ok1(z)) -> F3(x, y, z)

The TRS R consists of the following rules:

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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 4 SCCs with 5 less nodes.

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

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

F3(x, y, mark1(z)) -> F3(x, y, z)
F3(ok1(x), ok1(y), ok1(z)) -> F3(x, y, z)

The TRS R consists of the following rules:

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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.


F3(ok1(x), ok1(y), ok1(z)) -> F3(x, y, z)
The remaining pairs can at least be oriented weakly.

F3(x, y, mark1(z)) -> F3(x, y, z)
Used ordering: Polynomial interpretation [21]:

POL(F3(x1, x2, x3)) = 2·x2   
POL(mark1(x1)) = 0   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



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

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

F3(x, y, mark1(z)) -> F3(x, y, z)

The TRS R consists of the following rules:

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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.


F3(x, y, mark1(z)) -> F3(x, y, z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(F3(x1, x2, x3)) = 2·x3   
POL(mark1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



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

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

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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

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

PROPER1(f3(x, y, z)) -> PROPER1(x)
PROPER1(f3(x, y, z)) -> PROPER1(z)
PROPER1(f3(x, y, z)) -> PROPER1(y)

The TRS R consists of the following rules:

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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(f3(x, y, z)) -> PROPER1(x)
PROPER1(f3(x, y, z)) -> PROPER1(z)
PROPER1(f3(x, y, z)) -> PROPER1(y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = 2·x1   
POL(f3(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   

The following usable rules [14] were oriented: none



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

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

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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

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

ACTIVE1(f3(x, y, z)) -> ACTIVE1(z)

The TRS R consists of the following rules:

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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(f3(x, y, z)) -> ACTIVE1(z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = 2·x1   
POL(f3(x1, x2, x3)) = 1 + 2·x3   

The following usable rules [14] were oriented: none



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

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

active1(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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

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(f3(b, c, x)) -> mark1(f3(x, x, x))
active1(f3(x, y, z)) -> f3(x, y, active1(z))
active1(d) -> m1(b)
f3(x, y, mark1(z)) -> mark1(f3(x, y, z))
active1(d) -> mark1(c)
proper1(b) -> ok1(b)
proper1(c) -> ok1(c)
proper1(d) -> ok1(d)
proper1(f3(x, y, z)) -> f3(proper1(x), proper1(y), proper1(z))
f3(ok1(x), ok1(y), ok1(z)) -> ok1(f3(x, y, z))
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.