(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(c(x)))
c(b(x)) → a(c(a(x)))
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → c(b(a(x)))
b(c(x)) → a(c(a(x)))
Q is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x)) → B(a(x))
B(c(x)) → A(c(a(x)))
B(c(x)) → A(x)
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → c(b(a(x)))
b(c(x)) → a(c(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 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(x)) → A(x)
A(a(x)) → B(a(x))
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → c(b(a(x)))
b(c(x)) → a(c(a(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(a(x))
The graph contains the following edges 1 >= 1
- B(c(x)) → A(x)
The graph contains the following edges 1 > 1
(8) YES