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-453.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 (⇔, 11 ms)
↳8 QTRS
↳9 DependencyPairsProof (⇔, 10 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 QDP
↳13 QDPOrderProof (⇔, 44 ms)
↳14 QDP
↳15 DependencyGraphProof (⇔, 0 ms)
↳16 QDP
↳17 UsableRulesProof (⇔, 0 ms)
↳18 QDP
↳19 QDPOrderProof (⇔, 0 ms)
↳20 QDP
↳21 PisEmptyProof (⇔, 0 ms)
↳22 YES

(0) Obligation:

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

a(x) → b(c(x))
b(a(b(x))) → a(a(a(x)))
c(c(x)) → a(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) → c(b(x))
b(a(b(x))) → a(a(a(x)))
c(c(x)) → a(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(c(b(x)))
c(a(x)) → c(c(b(x)))
b(a(x)) → b(c(b(x)))
a(b(a(b(x)))) → a(a(a(a(x))))
c(b(a(b(x)))) → c(a(a(a(x))))
b(b(a(b(x)))) → b(a(a(a(x))))
a(c(c(x))) → a(a(x))
c(c(c(x))) → c(a(x))
b(c(c(x))) → b(a(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_{c_1}(c_{b_1}(b_{a_1}(x)))
a_{a_1}(a_{c_1}(x)) → a_{c_1}(c_{b_1}(b_{c_1}(x)))
a_{a_1}(a_{b_1}(x)) → a_{c_1}(c_{b_1}(b_{b_1}(x)))
c_{a_1}(a_{a_1}(x)) → c_{c_1}(c_{b_1}(b_{a_1}(x)))
c_{a_1}(a_{c_1}(x)) → c_{c_1}(c_{b_1}(b_{c_1}(x)))
c_{a_1}(a_{b_1}(x)) → c_{c_1}(c_{b_1}(b_{b_1}(x)))
b_{a_1}(a_{a_1}(x)) → b_{c_1}(c_{b_1}(b_{a_1}(x)))
b_{a_1}(a_{c_1}(x)) → b_{c_1}(c_{b_1}(b_{c_1}(x)))
b_{a_1}(a_{b_1}(x)) → b_{c_1}(c_{b_1}(b_{b_1}(x)))
a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{c_1}(c_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
c_{c_1}(c_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{c_1}(c_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))
c_{c_1}(c_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{c_1}(c_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = 1 + x1   
POL(a_{c_1}(x1)) = x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = 1 + x1   
POL(b_{c_1}(x1)) = 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_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{c_1}(c_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
c_{c_1}(c_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{c_1}(c_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{c_1}(c_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))


(8) Obligation:

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

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

Q is empty.

(9) DependencyPairsProof (EQUIVALENT transformation)

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

(10) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 20 less nodes.

(12) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

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

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

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

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.

(16) Obligation:

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

B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)

The TRS R consists of the following rules:

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

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

(17) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(18) Obligation:

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

B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)

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

(19) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(B_{A_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 1 + x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(20) Obligation:

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) YES