(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → x
a(b(x)) → b(c(x))
a(c(x)) → c(b(a(a(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(a(x))
A(c(x)) → A(x)
The TRS R consists of the following rules:
a(b(x)) → x
a(b(x)) → b(c(x))
a(c(x)) → c(b(a(a(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)) → 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 - 2} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(x)) → x
a(b(x)) → b(c(x))
a(c(x)) → c(b(a(a(x))))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(c(x)) → A(a(x))
The TRS R consists of the following rules:
a(b(x)) → x
a(b(x)) → b(c(x))
a(c(x)) → c(b(a(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.
A(c(x)) → A(a(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:
POL(A(x1)) = [4]x1
POL(a(x1)) = [2]x1
POL(b(x1)) = [2] + [1/2]x1
POL(c(x1)) = [4] + [2]x1
The value of delta used in the strict ordering is 16.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(x)) → x
a(b(x)) → b(c(x))
a(c(x)) → c(b(a(a(x))))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(x)) → x
a(b(x)) → b(c(x))
a(c(x)) → c(b(a(a(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) YES