(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → x
a(c(x)) → c(c(b(a(x))))
b(c(x)) → a(b(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:
b(a(x)) → x
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(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:
C(a(x)) → B(c(c(x)))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
C(b(x)) → B(a(x))
The TRS R consists of the following rules:
b(a(x)) → x
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(a(x)) → C(x)
C(a(x)) → C(c(x))
The TRS R consists of the following rules:
b(a(x)) → x
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
C(a(x)) → C(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( b(x1) ) = max{0, x1 - 2} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(a(x))
b(a(x)) → x
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(a(x)) → C(c(x))
The TRS R consists of the following rules:
b(a(x)) → x
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
C(a(x)) → C(c(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:
POL(C(x1)) = x1
POL(a(x1)) = [1] + [4]x1
POL(b(x1)) = [3/4] + [1/4]x1
POL(c(x1)) = [4]x1
The value of delta used in the strict ordering is 1.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(a(x))
b(a(x)) → x
(10) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(a(x)) → x
c(a(x)) → a(b(c(c(x))))
c(b(x)) → b(a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) YES