(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