(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(p(x)) → p(a(A(x)))
a(A(x)) → A(a(x))
p(A(A(x))) → a(p(x))
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(a(x)) → A(a(p(x)))
A(a(x)) → a(A(x))
A(A(p(x))) → p(a(x))
Q is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(a(x)) → A1(a(p(x)))
P(a(x)) → P(x)
A1(a(x)) → A1(x)
A1(A(p(x))) → P(a(x))
The TRS R consists of the following rules:
p(a(x)) → A(a(p(x)))
A(a(x)) → a(A(x))
A(A(p(x))) → p(a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
P(a(x)) → P(x)
A1(a(x)) → A1(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = 4 + x1
POL(A1(x1)) = 2·x1
POL(P(x1)) = 4·x1
POL(a(x1)) = 4 + x1
POL(p(x1)) = 4 + 2·x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
p(a(x)) → A(a(p(x)))
A(a(x)) → a(A(x))
A(A(p(x))) → p(a(x))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(a(x)) → A1(a(p(x)))
A1(A(p(x))) → P(a(x))
The TRS R consists of the following rules:
p(a(x)) → A(a(p(x)))
A(a(x)) → a(A(x))
A(A(p(x))) → p(a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(8) TRUE