Termination w.r.t. Q proof of /tmp/tmp_UQBw

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 UsableRulesProof (⇔)

    ↳7 QDP

    ↳8 UsableRulesReductionPairsProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 ForwardInstantiation (⇔)

    ↳16 QDP

    ↳17 RootLabelingFC2Proof (⇔)

    ↳18 QDP

    ↳19 DependencyGraphProof (⇔)

    ↳20 QDP

    ↳21 QDPOrderProof (⇔)

    ↳22 QDP

    ↳23 QDPOrderProof (⇔)

    ↳24 QDP

    ↳25 QDPOrderProof (⇔)

    ↳26 QDP

    ↳27 DependencyGraphProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 DependencyGraphProof (⇔)

    ↳32 QDP

    ↳33 QDPOrderProof (⇔)

    ↳34 QDP

    ↳35 PisEmptyProof (⇔)

    ↳36 TRUE

(0) Obligation:

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

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(4) Complex Obligation (AND)

(5) Obligation:

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

A(b(a(x0))) → A(b(x0))

The TRS R consists of the following rules:

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

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

(6) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(7) Obligation:

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

A(b(a(x0))) → A(b(x0))

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

(8) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

A(b(a(x0))) → A(b(x0))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(A(x₁)) = 2·x₁
POL(a(x₁)) = 2·x₁
POL(b(x₁)) = 2·x₁

(9) Obligation:

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(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(b(a(x1)), x2)) → P(a(b(a(x1))), x2)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

The following usable rules [FROCOS05] were oriented:

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

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule P(a(x0), p(b(a(x1)), x2)) → P(x1, p(a(b(a(x1))), x2)) we obtained the following new rules [LPAR04]:

P(a(x0), p(b(a(a(y_0))), x2)) → P(a(y_0), p(a(b(a(a(y_0)))), x2))

(16) Obligation:

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

P(a(x0), p(b(a(a(y_0))), x2)) → P(a(y_0), p(a(b(a(a(y_0)))), x2))

The TRS R consists of the following rules:

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

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

(17) 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.

(18) Obligation:

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

P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,a}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → P_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
P_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → P_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
P_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → P_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
P_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → P_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
P_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → P_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
P_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → P_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
P_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → P_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
P_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → P_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
P_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → P_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
P_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → P_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
P_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → P_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
P_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → P_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
P_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → P_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
P_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → P_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
P_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → P_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
P_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → P_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
P_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → P_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
P_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → P_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 27 less nodes.

(20) Obligation:

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

P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(21) 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_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), 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)) = x₁
POL(a_{b}(x₁)) = 0
POL(a_{p}(x₁)) = 1 + x₁
POL(b_{a}(x₁)) = x₁
POL(b_{b}(x₁)) = 0
POL(b_{p}(x₁)) = 0
POL(p_{a,a}(x₁, x₂)) = 1 + x₂
POL(p_{a,b}(x₁, x₂)) = 0
POL(p_{a,p}(x₁, x₂)) = 0
POL(p_{b,a}(x₁, x₂)) = 0
POL(p_{b,b}(x₁, x₂)) = x₁
POL(p_{b,p}(x₁, x₂)) = 0
POL(p_{p,p}(x₁, x₂)) = 0

The following usable rules [FROCOS05] were oriented:

p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)

(22) Obligation:

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

P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(23) 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_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,b}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), 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
POL(a_{b}(x₁)) = 0
POL(a_{p}(x₁)) = 0
POL(b_{a}(x₁)) = x₁
POL(b_{b}(x₁)) = 0
POL(b_{p}(x₁)) = 0
POL(p_{a,a}(x₁, x₂)) = 1 + x₂
POL(p_{a,b}(x₁, x₂)) = 0
POL(p_{a,p}(x₁, x₂)) = 0
POL(p_{b,a}(x₁, x₂)) = 0
POL(p_{b,b}(x₁, x₂)) = x₁
POL(p_{b,p}(x₁, x₂)) = 0
POL(p_{p,p}(x₁, x₂)) = 0

The following usable rules [FROCOS05] were oriented:

p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)

(24) Obligation:

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

P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(25) 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_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(P_{a,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(a_{p}(x₁)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(p_{b,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(b_{a}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(a_{a}(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(p_{a,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(a_{b}(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(p_{b,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{a,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{b,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(p_{a,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(p_{p,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{p}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(b_{b}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)

(26) Obligation:

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

P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{p}(y_0))), x2)) → P_{a,p}(a_{p}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{p}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(28) Obligation:

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

P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{a}(y_0))), x2)) → P_{a,p}(a_{a}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{a}(y_0)))), x2))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(P_{a,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(a_{a}(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(p_{b,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(b_{a}(x₁)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(p_{a,p}(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 1 0 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(a_{b}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(a_{p}(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁

POL(p_{b,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{a,b}(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{b,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{a,a}(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{p,p}(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{p}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(b_{b}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)

(30) Obligation:

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

P_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))
P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(31) DependencyGraphProof (EQUIVALENT transformation)

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

(32) Obligation:

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

P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The TRS R consists of the following rules:

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(a_{b}(y_0))), x2)) → P_{a,p}(a_{b}(y_0), p_{a,p}(a_{b}(b_{a}(a_{a}(a_{b}(y_0)))), x2))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(P_{a,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(a_{b}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(p_{b,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(b_{a}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(a_{a}(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(p_{a,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(a_{p}(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(p_{b,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{a,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{b,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{a,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(p_{p,p}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{p}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(b_{b}(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)

(34) Obligation:

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

p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{p}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{a}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{p}(x1)), x2)) → p_{p,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{p}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{a}(x1)), x2)) → p_{a,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{a}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,p}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,p}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,a}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,a}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(x0), p_{b,b}(b_{a}(a_{b}(x1)), x2)) → p_{b,p}(x1, p_{a,b}(a_{b}(b_{a}(a_{b}(x1))), x2))
p_{a,p}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{p}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{p}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{a}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{a}(_x0))), flat1)
p_{a,p}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,p}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,a}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,a}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{a,b}(a_{b}(b_{a}(a_{b}(_x0))), flat1) → p_{b,b}(b_{a}(a_{b}(b_{b}(_x0))), flat1)
p_{p,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{p,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{p,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{a,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{a,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{p}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{p}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{a}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{a}(_x0))))
p_{b,a}(flat0, a_{b}(b_{a}(a_{b}(_x0)))) → p_{b,b}(flat0, b_{a}(a_{b}(b_{b}(_x0))))
a_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{p}(_x0))))
a_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{a}(_x0))))
a_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → a_{b}(b_{a}(a_{b}(b_{b}(_x0))))
b_{a}(a_{b}(b_{a}(a_{p}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{p}(_x0))))
b_{a}(a_{b}(b_{a}(a_{a}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{a}(_x0))))
b_{a}(a_{b}(b_{a}(a_{b}(_x0)))) → b_{b}(b_{a}(a_{b}(b_{b}(_x0))))

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) UsableRulesProof (EQUIVALENT transformation)

(7) Obligation:

(8) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(9) Obligation:

(10) PisEmptyProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) ForwardInstantiation (EQUIVALENT transformation)

(16) Obligation:

(17) RootLabelingFC2Proof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) DependencyGraphProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) PisEmptyProof (EQUIVALENT transformation)

(36) TRUE