YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Mixed_SRS/touzet.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 (⇔, 12 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

(0) Obligation:

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

a(b(x)) → b(b(b(x)))
b(a(x)) → a(a(a(x)))
a(x) → x
b(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:

b(a(x)) → b(b(b(x)))
a(b(x)) → a(a(a(x)))
a(x) → x
b(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:

b(a(x)) → b(b(b(x)))
a(b(x)) → a(a(a(x)))
b(a(x)) → b(x)
a(a(x)) → a(x)
b(b(x)) → b(x)
a(b(x)) → 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:

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


(8) Obligation:

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

b_{a_1}(a_{a_1}(x)) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{a_1}(a_{a_1}(a_{b_1}(x)))
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
b_{b_1}(b_{b_1}(x)) → b_{b_1}(x)
b_{b_1}(b_{a_1}(x)) → b_{a_1}(x)
a_{b_1}(b_{b_1}(x)) → 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:

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

The TRS R consists of the following rules:

b_{a_1}(a_{a_1}(x)) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{a_1}(a_{a_1}(a_{b_1}(x)))
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
b_{b_1}(b_{b_1}(x)) → b_{b_1}(x)
b_{b_1}(b_{a_1}(x)) → b_{a_1}(x)
a_{b_1}(b_{b_1}(x)) → 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 2 SCCs with 4 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

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

A_{B_1}(b_{b_1}(x)) → A_{B_1}(x)

The TRS R consists of the following rules:

b_{a_1}(a_{a_1}(x)) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{a_1}(a_{a_1}(a_{b_1}(x)))
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
b_{b_1}(b_{b_1}(x)) → b_{b_1}(x)
b_{b_1}(b_{a_1}(x)) → b_{a_1}(x)
a_{b_1}(b_{b_1}(x)) → a_{b_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:

A_{B_1}(b_{b_1}(x)) → A_{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:

  • A_{B_1}(b_{b_1}(x)) → A_{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:

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