(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: Combined order from the following AFS and order.
P(x1, x2)  =  P(x2)
a(x1)  =  a
p(x1, x2)  =  p(x2)
b(x1)  =  b(x1)

Recursive Path Order [RPO].
Precedence:
a > p1 > P1
b1 > P1

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) 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(x2), p(a(a(b(x0))), x2))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(x1, x2)  =  P(x1, x2)
a(x1)  =  a(x1)
p(x1, x2)  =  p(x1, x2)
b(x1)  =  b

Recursive Path Order [RPO].
Precedence:
p2 > P2 > b
p2 > a1 > b

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))

(6) Obligation:

Q DP problem:
P is empty.
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.

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE