YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-30.srs

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 FlatCCProof (⇔, 0 ms)
↳4 QTRS
↳5 RootLabelingProof (⇔, 0 ms)
↳6 QTRS
↳7 QTRSRRRProof (⇔, 37 ms)
↳8 QTRS
↳9 QTRSRRRProof (⇔, 8 ms)
↳10 QTRS
↳11 DependencyPairsProof (⇔, 0 ms)
↳12 QDP
↳13 DependencyGraphProof (⇔, 0 ms)
↳14 QDP
↳15 QDPOrderProof (⇔, 91 ms)
↳16 QDP
↳17 DependencyGraphProof (⇔, 0 ms)
↳18 TRUE

(0) Obligation:

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

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

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

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

Q is empty.

(3) FlatCCProof (EQUIVALENT transformation)

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

(4) Obligation:

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

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

Q is empty.

(5) RootLabelingProof (EQUIVALENT transformation)

We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(6) Obligation:

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

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

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = 2 + x1   
POL(a_{b_1}(x1)) = 1 + x1   
POL(a_{c_1}(x1)) = x1   
POL(b_{a_1}(x1)) = 3 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = 1 + x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{b_1}(x1)) = x1   
POL(c_{c_1}(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(8) Obligation:

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

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

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = 2 + x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = x1   
POL(b_{a_1}(x1)) = 3 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = 1 + x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{b_1}(x1)) = x1   
POL(c_{c_1}(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(10) Obligation:

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

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

Q is empty.

(11) DependencyPairsProof (EQUIVALENT transformation)

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

(12) Obligation:

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

C_{A_1}(a_{a_1}(x)) → C_{B_1}(b_{b_1}(b_{a_1}(x)))
C_{A_1}(a_{a_1}(x)) → B_{B_1}(b_{a_1}(x))
C_{A_1}(a_{c_1}(x)) → C_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{A_1}(a_{c_1}(x)) → B_{B_1}(b_{c_1}(x))
C_{A_1}(a_{c_1}(x)) → B_{C_1}(x)
B_{B_1}(b_{a_1}(a_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x)))))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x))))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{C_1}(c_{a_1}(a_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x)))))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x))))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{C_1}(c_{a_1}(a_{b_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(x))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x)))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → A_{C_1}(c_{a_1}(a_{a_1}(x)))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x))))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → A_{C_1}(c_{a_1}(a_{b_1}(x)))
C_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{b_1}(x))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → A_{C_1}(c_{a_1}(a_{c_1}(x)))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(x))

The TRS R consists of the following rules:

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

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes.

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) 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}(x)) → C_{B_1}(b_{b_1}(b_{a_1}(x)))
C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{a_1}(x)) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{A_1}(a_{c_1}(x)) → C_{B_1}(b_{b_1}(b_{c_1}(x)))
C_{A_1}(a_{c_1}(x)) → B_{B_1}(b_{c_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x))))
B_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{B_1}(b_{a_1}(a_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( C_{A_1}(x1) ) = 2x1 + 1

POL( a_{c_1}(x1) ) = x1

POL( C_{B_1}(x1) ) = max{0, 2x1 - 1}

POL( b_{c_1}(x1) ) = x1

POL( c_{a_1}(x1) ) = x1 + 1

POL( a_{a_1}(x1) ) = x1 + 2

POL( c_{b_1}(x1) ) = x1 + 1

POL( b_{b_1}(x1) ) = x1

POL( b_{a_1}(x1) ) = x1 + 2

POL( c_{c_1}(x1) ) = x1 + 1

POL( a_{b_1}(x1) ) = 1

POL( B_{B_1}(x1) ) = 2x1


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

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

(16) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE