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-452.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 (⇔, 26 ms)
↳8 QTRS
↳9 DependencyPairsProof (⇔, 16 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 QDP
↳13 QDPOrderProof (⇔, 43 ms)
↳14 QDP
↳15 QDPOrderProof (⇔, 233 ms)
↳16 QDP
↳17 DependencyGraphProof (⇔, 0 ms)
↳18 QDP
↳19 QDPOrderProof (⇔, 18 ms)
↳20 QDP
↳21 QDPOrderProof (⇔, 668 ms)
↳22 QDP
↳23 DependencyGraphProof (⇔, 0 ms)
↳24 TRUE

(0) Obligation:

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

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

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = 1 + x1   
POL(a_{c_1}(x1)) = 2 + x1   
POL(b_{a_1}(x1)) = 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)) = 2 + 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}(x)
a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → a_{c_1}(x)
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → c_{b_1}(x)
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(x)
b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → b_{c_1}(x)
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(x)
a_{c_1}(c_{c_1}(c_{a_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))
a_{c_1}(c_{c_1}(c_{c_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(x))) → c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{c_1}(x))) → c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{b_1}(x))) → c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))
b_{c_1}(c_{c_1}(c_{c_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{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}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{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_{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_{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_{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}(x)
A_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{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_{a_1}(a_{b_1}(x)))
A_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{b_1}(x))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{a_1}(a_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → 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}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{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 10 less nodes.

(12) Obligation:

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

A_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{A_1}(a_{a_1}(x)) → B_{C_1}(c_{b_1}(b_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_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)))) → C_{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)
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_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{a_1}(a_{b_1}(x)))
A_{A_1}(a_{a_1}(x)) → A_{C_1}(c_{b_1}(b_{a_1}(x)))
A_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{a_1}(a_{b_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)
B_{A_1}(a_{c_1}(x)) → B_{C_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)) → B_{C_1}(x)
C_{A_1}(a_{c_1}(x)) → B_{C_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}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{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_{c_1}(x)) → A_{C_1}(c_{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   
POL(A_{C_1}(x1)) = x1   
POL(B_{A_1}(x1)) = 1   
POL(B_{C_1}(x1)) = 1   
POL(C_{A_1}(x1)) = 1   
POL(C_{B_1}(x1)) = 1   
POL(a_{a_1}(x1)) = 0   
POL(a_{b_1}(x1)) = 0   
POL(a_{c_1}(x1)) = 0   
POL(b_{a_1}(x1)) = 1   
POL(b_{b_1}(x1)) = 0   
POL(b_{c_1}(x1)) = 0   
POL(c_{a_1}(x1)) = 1   
POL(c_{b_1}(x1)) = x1   
POL(c_{c_1}(x1)) = 1   

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

(14) Obligation:

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

A_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{A_1}(a_{a_1}(x)) → B_{C_1}(c_{b_1}(b_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_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)))) → C_{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)
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_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{a_1}(a_{b_1}(x)))
A_{A_1}(a_{a_1}(x)) → A_{C_1}(c_{b_1}(b_{a_1}(x)))
A_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{a_1}(a_{b_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)
B_{A_1}(a_{c_1}(x)) → B_{C_1}(x)
A_{A_1}(a_{c_1}(x)) → B_{C_1}(x)
C_{A_1}(a_{c_1}(x)) → B_{C_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}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_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_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → C_{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)
B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)
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)
B_{A_1}(a_{c_1}(x)) → B_{C_1}(x)
A_{A_1}(a_{c_1}(x)) → B_{C_1}(x)
C_{A_1}(a_{c_1}(x)) → B_{C_1}(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A_{A_1}(x1) ) = x1 + 1

POL( B_{A_1}(x1) ) = x1

POL( C_{B_1}(x1) ) = x1

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

POL( b_{b_1}(x1) ) = max{0, x1 - 2}

POL( A_{C_1}(x1) ) = x1

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

POL( B_{C_1}(x1) ) = x1

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

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

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

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

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

POL( c_{c_1}(x1) ) = x1 + 2

POL( C_{A_1}(x1) ) = x1 + 1


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

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

(16) Obligation:

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

A_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{A_1}(a_{a_1}(x)) → B_{C_1}(c_{b_1}(b_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{A_1}(a_{a_1}(x)) → 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_{C_1}(c_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{a_1}(a_{b_1}(x)))
A_{C_1}(c_{c_1}(c_{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}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_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 1 SCC with 4 less nodes.

(18) Obligation:

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

B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{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)))

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

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_{c_1}(x)) → B_{C_1}(c_{b_1}(b_{c_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{A_1}(x1)) = 1   
POL(B_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 0   
POL(a_{b_1}(x1)) = 0   
POL(a_{c_1}(x1)) = 0   
POL(b_{a_1}(x1)) = 1   
POL(b_{b_1}(x1)) = 0   
POL(b_{c_1}(x1)) = 0   
POL(c_{a_1}(x1)) = 1   
POL(c_{b_1}(x1)) = x1   
POL(c_{c_1}(x1)) = 1   

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

(20) Obligation:

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

B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{A_1}(a_{a_1}(x)) → B_{C_1}(c_{b_1}(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}(x)
c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))) → c_{c_1}(x)
a_{c_1}(c_{c_1}(c_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_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.


B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(B_{C_1}(x1)) = -I +
[0A,0A,0A]
·x1

POL(c_{c_1}(x1)) =
/0A\
|0A|
\1A/
 +
/-I-I-I\
|-I0A-I|
\-I1A-I/
·x1

POL(c_{b_1}(x1)) =
/-I\
|-I|
\-I/
 +
/0A0A0A\
|0A0A-I|
\-I1A-I/
·x1

POL(B_{A_1}(x1)) = -I +
[0A,1A,-I]
·x1

POL(a_{a_1}(x1)) =
/0A\
|-I|
\-I/
 +
/0A0A0A\
|-I-I0A|
\-I1A-I/
·x1

POL(a_{b_1}(x1)) =
/-I\
|-I|
\0A/
 +
/0A-I-I\
|-I-I-I|
\0A0A-I/
·x1

POL(b_{a_1}(x1)) =
/-I\
|-I|
\0A/
 +
/0A-I-I\
|-I-I0A|
\0A0A0A/
·x1

POL(a_{c_1}(x1)) =
/0A\
|-I|
\-I/
 +
/0A0A-I\
|-I-I-I|
\-I-I0A/
·x1

POL(b_{b_1}(x1)) =
/-I\
|-I|
\-I/
 +
/0A-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(b_{c_1}(x1)) =
/-I\
|-I|
\0A/
 +
/0A0A-I\
|-I-I-I|
\0A0A-I/
·x1

POL(c_{a_1}(x1)) =
/0A\
|0A|
\1A/
 +
/0A-I-I\
|0A-I0A|
\1A-I-I/
·x1

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

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

(22) Obligation:

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

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

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE