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-271.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 (⇔, 33 ms)
↳8 QTRS
↳9 DependencyPairsProof (⇔, 0 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 AND
    ↳13 QDP
        ↳14 UsableRulesProof (⇔, 0 ms)
        ↳15 QDP
        ↳16 QDPSizeChangeProof (⇔, 0 ms)
        ↳17 YES
    ↳18 QDP
        ↳19 UsableRulesProof (⇔, 0 ms)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔, 0 ms)
        ↳22 YES
    ↳23 QDP
        ↳24 UsableRulesProof (⇔, 0 ms)
        ↳25 QDP
        ↳26 QDPOrderProof (⇔, 57 ms)
        ↳27 QDP
        ↳28 PisEmptyProof (⇔, 0 ms)
        ↳29 YES

(0) Obligation:

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

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

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

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

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)
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_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{a_1}(a_{c_1}(c_{a_1}(x)))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{a_1}(a_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{a_1}(a_{c_1}(c_{c_1}(x)))
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{a_1}(a_{c_1}(c_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{a_1}(a_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{a_1}(a_{c_1}(c_{c_1}(x)))
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{a_1}(a_{c_1}(c_{a_1}(x)))
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{a_1}(a_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{a_1}(a_{c_1}(c_{c_1}(x)))
b_{c_1}(c_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x))))))
b_{c_1}(c_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x))))))
b_{c_1}(c_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(c_{c_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 3 SCCs with 23 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

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

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

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

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

(14) 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.

(15) Obligation:

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

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

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

(16) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • C_{B_1}(b_{a_1}(a_{a_1}(x))) → C_{A_1}(x)
    The graph contains the following edges 1 > 1

  • C_{B_1}(b_{a_1}(a_{b_1}(x))) → C_{B_1}(x)
    The graph contains the following edges 1 > 1

  • C_{A_1}(a_{a_1}(x)) → C_{A_1}(x)
    The graph contains the following edges 1 > 1

  • C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)
    The graph contains the following edges 1 > 1

(17) YES

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

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

(19) 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.

(20) 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.

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)
    The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

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

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

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

(24) 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.

(25) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(26) 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_{a_1}(x))) → B_{C_1}(c_{a_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(x))) → B_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{b_1}(x))) → B_{C_1}(c_{b_1}(x))
B_{C_1}(c_{c_1}(c_{c_1}(x))) → B_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{c_1}(x))) → B_{C_1}(c_{c_1}(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = 1 + 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)) = 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:

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

(27) Obligation:

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

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

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) YES