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