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