(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(p(b(a(x0)), x1), p(x2, x3)) → p(p(x3, a(x2)), p(b(a(x1)), 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(p(b(a(x0)), x1), p(x2, x3)) → P(p(x3, a(x2)), p(b(a(x1)), b(x0)))
P(p(b(a(x0)), x1), p(x2, x3)) → P(x3, a(x2))
P(p(b(a(x0)), x1), p(x2, x3)) → P(b(a(x1)), b(x0))
The TRS R consists of the following rules:
p(p(b(a(x0)), x1), p(x2, x3)) → p(p(x3, a(x2)), p(b(a(x1)), 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 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(p(b(a(x0)), x1), p(x2, x3)) → P(p(x3, a(x2)), p(b(a(x1)), b(x0)))
The TRS R consists of the following rules:
p(p(b(a(x0)), x1), p(x2, x3)) → p(p(x3, a(x2)), p(b(a(x1)), b(x0)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) 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.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(p(b(a(x0)), x1), p(x2, x3)) → P(p(x3, a(x2)), p(b(a(x1)), b(x0)))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
P(p(b(a(x0)), x1), p(x2, x3)) → P(p(x3, a(x2)), p(b(a(x1)), b(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(P(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(p(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
none
(8) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE