(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