(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) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
C(b(a(X))) → C(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
C(
x0,
x1) =
C(
x1)
Tags:
C has argument tags [0,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(C(x1)) = 0
POL(a(x1)) = 1 + x1
POL(b(x1)) = x1
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
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.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE