(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(b(a(X))) → a(a(b(b(c(c(X))))))
a(X) → e
b(X) → e
c(X) → e
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:
C(b(a(X))) → A(a(b(b(c(c(X))))))
C(b(a(X))) → A(b(b(c(c(X)))))
C(b(a(X))) → B(b(c(c(X))))
C(b(a(X))) → B(c(c(X)))
C(b(a(X))) → C(c(X))
C(b(a(X))) → C(X)
The TRS R consists of the following rules:
c(b(a(X))) → a(a(b(b(c(c(X))))))
a(X) → e
b(X) → e
c(X) → e
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 5 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(b(a(X))) → C(X)
The TRS R consists of the following rules:
c(b(a(X))) → a(a(b(b(c(c(X))))))
a(X) → e
b(X) → e
c(X) → e
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
a(x1) = x1
b(x1) = b(x1)
From the DPs we obtained the following set of size-change graphs:
- C(b(a(X))) → C(X) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(6) TRUE