(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
a(b(x)) → b(b(a(c(x))))
b(b(x)) → a(x)
c(c(x)) → 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
b(a(x)) → c(a(b(b(x))))
b(b(x)) → a(x)
c(c(x)) → x
Q is empty.
(3) FlatCCProof (EQUIVALENT transformation)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → a(x)
b(a(x)) → b(x)
c(a(x)) → c(x)
a(b(a(x))) → a(c(a(b(b(x)))))
b(b(a(x))) → b(c(a(b(b(x)))))
c(b(a(x))) → c(c(a(b(b(x)))))
a(b(b(x))) → a(a(x))
b(b(b(x))) → b(a(x))
c(b(b(x))) → c(a(x))
a(c(c(x))) → a(x)
b(c(c(x))) → b(x)
c(c(c(x))) → c(x)
Q is empty.
(5) RootLabelingProof (EQUIVALENT transformation)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{b_1}(b_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{b_1}(b_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{b_1}(b_{c_1}(x))) → c_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(x)
a_{c_1}(c_{c_1}(c_{c_1}(x))) → a_{c_1}(x)
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(x)
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(x)
b_{c_1}(c_{c_1}(c_{c_1}(x))) → b_{c_1}(x)
c_{c_1}(c_{c_1}(c_{a_1}(x))) → c_{a_1}(x)
c_{c_1}(c_{c_1}(c_{b_1}(x))) → c_{b_1}(x)
c_{c_1}(c_{c_1}(c_{c_1}(x))) → c_{c_1}(x)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a_{a_1}(x1)) = 2 + x1
POL(a_{b_1}(x1)) = x1
POL(a_{c_1}(x1)) = 1 + x1
POL(b_{a_1}(x1)) = 3 + x1
POL(b_{b_1}(x1)) = 1 + x1
POL(b_{c_1}(x1)) = 2 + x1
POL(c_{a_1}(x1)) = x1
POL(c_{b_1}(x1)) = x1
POL(c_{c_1}(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{b_1}(b_{b_1}(b_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{b_1}(b_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{b_1}(b_{c_1}(x))) → c_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(x)
a_{c_1}(c_{c_1}(c_{c_1}(x))) → a_{c_1}(x)
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(x)
b_{c_1}(c_{c_1}(c_{c_1}(x))) → b_{c_1}(x)
c_{c_1}(c_{c_1}(c_{a_1}(x))) → c_{a_1}(x)
c_{c_1}(c_{c_1}(c_{b_1}(x))) → c_{b_1}(x)
c_{c_1}(c_{c_1}(c_{c_1}(x))) → c_{c_1}(x)
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(x)
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(x)
Q is empty.
(9) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)
A_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
A_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{C_1}(x)
B_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{C_1}(x)
C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
C_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{C_1}(x)
A_{B_1}(b_{b_1}(b_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(x))) → A_{C_1}(x)
B_{B_1}(b_{b_1}(b_{b_1}(x))) → A_{B_1}(x)
B_{B_1}(b_{b_1}(b_{c_1}(x))) → A_{C_1}(x)
The TRS R consists of the following rules:
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(x)
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{b_1}(b_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{b_1}(b_{b_1}(x))) → A_{B_1}(x)
C_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
C_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
The TRS R consists of the following rules:
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(x)
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{b_1}(b_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{b_1}(b_{b_1}(x))) → A_{B_1}(x)
C_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{a_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
C_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A_{B_1}(x1) ) = max{0, x1 - 1} |
POL( C_{A_1}(x1) ) = max{0, x1 - 1} |
POL( a_{b_1}(x1) ) = x1 + 1 |
POL( b_{b_1}(x1) ) = x1 + 1 |
POL( b_{a_1}(x1) ) = x1 + 2 |
POL( a_{a_1}(x1) ) = x1 + 2 |
POL( c_{a_1}(x1) ) = x1 + 1 |
POL( c_{c_1}(x1) ) = x1 + 1 |
POL( c_{b_1}(x1) ) = x1 + 2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(x)
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)
A_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
The TRS R consists of the following rules:
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(x)
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
(16) TRUE