(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
P(
x0,
x1,
x2) =
P(
x2)
Tags:
P has argument tags [0,2,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
P(
x1,
x2) =
x1
a(
x1) =
a(
x1)
p(
x1,
x2) =
p(
x2)
b(
x1) =
b(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
[a1, p1] > b1
Status:
a1: multiset
p1: multiset
b1: multiset
The following usable rules [FROCOS05] were oriented:
none
(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
P(
x0,
x1,
x2) =
P(
x2)
Tags:
P has argument tags [1,0,1] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
P(
x1,
x2) =
x2
a(
x1) =
a(
x1)
p(
x1,
x2) =
p(
x1,
x2)
b(
x1) =
b
Recursive path order with status [RPO].
Quasi-Precedence:
p2 > [a1, b]
Status:
a1: multiset
p2: multiset
b: multiset
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