Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
  ↳3 QDPOrderProof (⇔)
    ↳4 QDP
    ↳5 RootLabelingFC2Proof (⇔)
    ↳6 QDP
    ↳7 DependencyGraphProof (⇔)
    ↳8 QDP
    ↳9 QDPOrderProof (⇔)
    ↳10 QDP
    ↳11 PisEmptyProof (⇔)
    ↳12 TRUE
  ↳13 QDPOrderProof (⇔)
    ↳14 QDP

(0) Obligation:

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

p(a(x0), p(a(a(a(x1))), x2)) → p(a(x2), p(a(a(b(x0))), x2))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

P(a(x0), p(a(a(a(x1))), x2)) → P(a(x2), p(a(a(b(x0))), x2))
P(a(x0), p(a(a(a(x1))), x2)) → P(a(a(b(x0))), x2)

The TRS R consists of the following rules:

p(a(x0), p(a(a(a(x1))), x2)) → p(a(x2), p(a(a(b(x0))), x2))

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P(a(x0), p(a(a(a(x1))), x2)) → P(a(a(b(x0))), x2)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(P(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(a(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(p(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(b(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

The following usable rules [FROCOS05] were oriented:

p(a(x0), p(a(a(a(x1))), x2)) → p(a(x2), p(a(a(b(x0))), x2))

(4) Obligation:

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

P(a(x0), p(a(a(a(x1))), x2)) → P(a(x2), p(a(a(b(x0))), x2))

The TRS R consists of the following rules:

p(a(x0), p(a(a(a(x1))), x2)) → p(a(x2), p(a(a(b(x0))), x2))

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

(5) RootLabelingFC2Proof (EQUIVALENT transformation)

We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(6) Obligation:

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

P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
P_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 24 less nodes.

(8) Obligation:

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

P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
P_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → P_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(P_{a,p}(x1, x2)) = x2   
POL(a_{a}(x1)) = 1 + x1   
POL(a_{b}(x1)) = 0   
POL(a_{p}(x1)) = 0   
POL(b_{a}(x1)) = 0   
POL(b_{p}(x1)) = 0   
POL(p_{a,a}(x1, x2)) = x1 + x1·x2   
POL(p_{a,b}(x1, x2)) = x1·x2   
POL(p_{a,p}(x1, x2)) = x1 + x2   

The following usable rules [FROCOS05] were oriented:

p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))

(10) Obligation:

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

p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{p}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{p}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{a}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{a}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{p}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{a}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,p}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{p}(x2), p_{a,p}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,a}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{a}(x2), p_{a,a}(a_{a}(a_{b}(b_{b}(x0))), x2))
p_{a,p}(a_{b}(x0), p_{a,b}(a_{a}(a_{a}(a_{b}(x1))), x2)) → p_{a,p}(a_{b}(x2), p_{a,b}(a_{a}(a_{b}(b_{b}(x0))), x2))

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

(11) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(12) TRUE

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P(a(x0), p(a(a(a(x1))), x2)) → P(a(a(b(x0))), x2)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(P(x1, x2)) = x2   
POL(a(x1)) = 0   
POL(b(x1)) = 0   
POL(p(x1, x2)) = 1 + x2   

The following usable rules [FROCOS05] were oriented:

p(a(x0), p(a(a(a(x1))), x2)) → p(a(x2), p(a(a(b(x0))), x2))

(14) Obligation:

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

P(a(x0), p(a(a(a(x1))), x2)) → P(a(x2), p(a(a(b(x0))), x2))

The TRS R consists of the following rules:

p(a(x0), p(a(a(a(x1))), x2)) → p(a(x2), p(a(a(b(x0))), x2))

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