(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(x))) → a(b(c(x)))
c(a(x)) → a(c(x))
c(b(x)) → b(a(x))
a(a(x)) → a(b(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:
B(a(a(x))) → A(b(c(x)))
B(a(a(x))) → B(c(x))
B(a(a(x))) → C(x)
C(a(x)) → A(c(x))
C(a(x)) → C(x)
C(b(x)) → B(a(x))
C(b(x)) → A(x)
A(a(x)) → A(b(a(x)))
A(a(x)) → B(a(x))
The TRS R consists of the following rules:
b(a(a(x))) → a(b(c(x)))
c(a(x)) → a(c(x))
c(b(x)) → b(a(x))
a(a(x)) → a(b(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.
B(a(a(x))) → A(b(c(x)))
B(a(a(x))) → B(c(x))
B(a(a(x))) → C(x)
C(a(x)) → A(c(x))
C(a(x)) → C(x)
C(b(x)) → A(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(B(x1)) = x1
POL(C(x1)) = 1 + x1
POL(a(x1)) = 1 + x1
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(a(x)) → a(c(x))
c(b(x)) → b(a(x))
b(a(a(x))) → a(b(c(x)))
a(a(x)) → a(b(a(x)))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(b(x)) → B(a(x))
A(a(x)) → A(b(a(x)))
A(a(x)) → B(a(x))
The TRS R consists of the following rules:
b(a(a(x))) → a(b(c(x)))
c(a(x)) → a(c(x))
c(b(x)) → b(a(x))
a(a(x)) → a(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:
A(a(x)) → A(b(a(x)))
The TRS R consists of the following rules:
b(a(a(x))) → a(b(c(x)))
c(a(x)) → a(c(x))
c(b(x)) → b(a(x))
a(a(x)) → a(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.
A(a(x)) → A(b(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( A(x1) ) = max{0, x1 - 1} |
| 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(a(x)) → a(b(a(x)))
b(a(a(x))) → a(b(c(x)))
c(a(x)) → a(c(x))
c(b(x)) → b(a(x))
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(a(a(x))) → a(b(c(x)))
c(a(x)) → a(c(x))
c(b(x)) → b(a(x))
a(a(x)) → a(b(a(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