(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → x
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:
A(c(x)) → A(b(b(c(c(a(x))))))
A(c(x)) → B(b(c(c(a(x)))))
A(c(x)) → B(c(c(a(x))))
A(c(x)) → A(x)
The TRS R consists of the following rules:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(c(x)) → A(x)
A(c(x)) → A(b(b(c(c(a(x))))))
The TRS R consists of the following rules:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → 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.
A(c(x)) → A(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( b(x1) ) = max{0, x1 - 1} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → x
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(c(x)) → A(b(b(c(c(a(x))))))
The TRS R consists of the following rules:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → 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.
A(c(x)) → A(b(b(c(c(a(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:
POL(A(x1)) = [1/4]x1
POL(a(x1)) = [4]x1
POL(b(x1)) = [1/4]x1
POL(c(x1)) = [1/2] + [4]x1
The value of delta used in the strict ordering is 11/128.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → x
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(x)) → x
a(c(x)) → a(b(b(c(c(a(x))))))
b(c(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) YES