(0) Obligation:

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

p(a(x0), p(x1, p(x2, x3))) → p(x1, p(x0, p(a(x3), x3)))

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(x1, p(x2, x3))) → P(x1, p(x0, p(a(x3), x3)))
P(a(x0), p(x1, p(x2, x3))) → P(x0, p(a(x3), x3))
P(a(x0), p(x1, p(x2, x3))) → P(a(x3), x3)

The TRS R consists of the following rules:

p(a(x0), p(x1, p(x2, x3))) → p(x1, p(x0, p(a(x3), x3)))

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(x1, p(x2, x3))) → P(x0, p(a(x3), x3))
P(a(x0), p(x1, p(x2, x3))) → P(a(x3), x3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(x1, x2)  =  x2
a(x1)  =  x1
p(x1, x2)  =  p(x2)

Lexicographic path order with status [LPO].
Precedence:
trivial

Status:
p1: [1]

The following usable rules [FROCOS05] were oriented:

p(a(x0), p(x1, p(x2, x3))) → p(x1, p(x0, p(a(x3), x3)))

(4) Obligation:

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

P(a(x0), p(x1, p(x2, x3))) → P(x1, p(x0, p(a(x3), x3)))

The TRS R consists of the following rules:

p(a(x0), p(x1, p(x2, x3))) → p(x1, p(x0, p(a(x3), x3)))

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