YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/dup08.srs

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a(a(a(a(b(b(x)))))) → b(b(a(a(b(b(x))))))
b(b(a(a(x)))) → a(a(b(b(b(b(x))))))
b(b(c(c(a(a(x)))))) → c(c(c(c(a(a(a(a(b(b(x))))))))))

Q is empty.

(1) FlatCCProof (EQUIVALENT transformation)

We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete.

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a(a(a(a(a(b(b(x))))))) → a(b(b(a(a(b(b(x)))))))
b(a(a(a(a(b(b(x))))))) → b(b(b(a(a(b(b(x)))))))
c(a(a(a(a(b(b(x))))))) → c(b(b(a(a(b(b(x)))))))
a(b(b(a(a(x))))) → a(a(a(b(b(b(b(x)))))))
b(b(b(a(a(x))))) → b(a(a(b(b(b(b(x)))))))
c(b(b(a(a(x))))) → c(a(a(b(b(b(b(x)))))))
a(b(b(c(c(a(a(x))))))) → a(c(c(c(c(a(a(a(a(b(b(x)))))))))))
b(b(b(c(c(a(a(x))))))) → b(c(c(c(c(a(a(a(a(b(b(x)))))))))))
c(b(b(c(c(a(a(x))))))) → c(c(c(c(c(a(a(a(a(b(b(x)))))))))))

Q is empty.

(3) 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.

(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{b_1}(x1)) = 1 + x1   
POL(c_{c_1}(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))


(6) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{B_1}(b_{a_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{A_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{A_1}(x)
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{A_1}(x)
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{B_1}(b_{a_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{A_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{A_1}(x)
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(x)
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{B_1}(b_{a_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → B_{A_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{A_1}(x1)) = x1   
POL(A_{B_1}(x1)) = x1   
POL(B_{A_1}(x1)) = x1   
POL(B_{B_1}(x1)) = x1   
POL(C_{A_1}(x1)) = 1 + x1   
POL(C_{B_1}(x1)) = 1 + x1   
POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{b_1}(x1)) = 1 + x1   
POL(c_{c_1}(x1)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.

(14) Complex Obligation (AND)

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(16) 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(C_{A_1}(x1)) = x1   
POL(C_{B_1}(x1)) = 1 + x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = 0   
POL(b_{a_1}(x1)) = 1 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = 0   
POL(c_{a_1}(x1)) = 0   
POL(c_{b_1}(x1)) = 0   
POL(c_{c_1}(x1)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(19) TRUE

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{A_1}(x1)) = 1 + x1   
POL(A_{B_1}(x1)) = 1 + x1   
POL(B_{A_1}(x1)) = x1   
POL(B_{B_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = 0   
POL(b_{a_1}(x1)) = 1 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = 0   
POL(c_{a_1}(x1)) = 0   
POL(c_{b_1}(x1)) = 1   
POL(c_{c_1}(x1)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

(22) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))))
b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(24) YES