(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(c(a(x))))
c(b(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:
A(a(x)) → A(b(c(a(x))))
A(a(x)) → C(a(x))
C(b(x)) → A(b(a(x)))
C(b(x)) → A(x)
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(c(a(x))))
c(b(x)) → a(b(a(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(a(x)) → C(a(x))
C(b(x)) → A(x)
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(c(a(x))))
c(b(x)) → a(b(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) 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:
- C(b(x)) → A(x)
The graph contains the following edges 1 > 1
- A(a(x)) → C(a(x))
The graph contains the following edges 1 >= 1
(6) YES