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-59.srs

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

0 QTRS
↳1 FlatCCProof (⇔, 0 ms)
↳2 QTRS
↳3 RootLabelingProof (⇔, 0 ms)
↳4 QTRS
↳5 QTRSRRRProof (⇔, 5 ms)
↳6 QTRS
↳7 DependencyPairsProof (⇔, 6 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 31 ms)
↳12 QDP
↳13 DependencyGraphProof (⇔, 0 ms)
↳14 TRUE

(0) Obligation:

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

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

Q is empty.

(5) 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)) = 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)) = 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:

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


(6) Obligation:

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

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

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x)))))
a_{a_1}(a_{b_1}(x)) → a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x)))))
a_{a_1}(a_{c_1}(x)) → a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_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_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{c_1}(c_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{c_1}(c_{a_1}(x)))
a_{c_1}(c_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{c_1}(c_{b_1}(x)))
a_{c_1}(c_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(c_{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 8 less nodes.

(10) Obligation:

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

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

The TRS R consists of the following rules:

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


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

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

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

(12) Obligation:

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

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

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x)))))
a_{a_1}(a_{b_1}(x)) → a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x)))))
a_{a_1}(a_{c_1}(x)) → a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(a_{c_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_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{c_1}(c_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{c_1}(c_{a_1}(x)))
a_{c_1}(c_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{c_1}(c_{b_1}(x)))
a_{c_1}(c_{b_1}(b_{c_1}(x))) → a_{a_1}(a_{c_1}(c_{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 0 SCCs with 3 less nodes.

(14) TRUE