(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(a(x))))
b(a(x)) → a(a(x))
a(a(x)) → a(c(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(b(x)) → B(a(a(a(x))))
A(b(x)) → A(a(a(x)))
A(b(x)) → A(a(x))
A(b(x)) → A(x)
B(a(x)) → A(a(x))
A(a(x)) → A(c(b(x)))
A(a(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(a(x))))
b(a(x)) → a(a(x))
a(a(x)) → a(c(b(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 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(x)) → A(a(x))
A(b(x)) → B(a(a(a(x))))
A(b(x)) → A(a(a(x)))
A(b(x)) → A(a(x))
A(b(x)) → A(x)
A(a(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(a(x))))
b(a(x)) → a(a(x))
a(a(x)) → a(c(b(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(b(x)) → B(a(a(a(x))))
A(b(x)) → A(a(a(x)))
A(b(x)) → A(a(x))
A(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(a(x1)) = x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = 0
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(a(x)) → a(a(x))
a(b(x)) → b(a(a(a(x))))
a(a(x)) → a(c(b(x)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(x)) → A(a(x))
A(a(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(a(x))))
b(a(x)) → a(a(x))
a(a(x)) → a(c(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- A(a(x)) → B(x)
The graph contains the following edges 1 > 1
- B(a(x)) → A(a(x))
The graph contains the following edges 1 >= 1
(8) YES