(0) Obligation:

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

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

The TRS R consists of the following rules:

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

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 1 SCC with 1 less node.

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

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

Lexicographic Path Order [LPO].
Precedence:
P1 > p1 > b1


The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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