(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) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A(b(a(x0))) → A(b(x0))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A(
x1) =
x1
b(
x1) =
x1
a(
x1) =
a(
x1)
Lexicographic Path Order [LPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
none
(7) Obligation:
Q DP problem:
P is empty.
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.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) 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.
(11) 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: Combined order from the following AFS and order.
P(
x1,
x2) =
x2
p(
x1,
x2) =
p(
x2)
Lexicographic Path Order [LPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
p(a(x0), p(b(a(x1)), x2)) → p(x1, p(a(b(a(x1))), x2))
(12) 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.