YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 16 ms]
        (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 106 ms]
        (13) QDP
        (14) PisEmptyProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 115 ms]
        (18) QDP
        (19) PisEmptyProof [EQUIVALENT, 0 ms]
        (20) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(b(a(b(x1)))) -> b(a(b(b(b(b(x1))))))
   b(a(b(b(x1)))) -> b(b(b(x1)))
   b(b(b(x1))) -> b(b(a(a(b(a(b(x1)))))))

Q is empty.

----------------------------------------

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(a(b(b(x1)))) -> b(b(b(b(a(b(x1))))))
   b(b(a(b(x1)))) -> b(b(b(x1)))
   b(b(b(x1))) -> b(a(b(a(a(b(b(x1)))))))

Q is empty.

----------------------------------------

(3) RootLabelingProof (EQUIVALENT)
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:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

Q is empty.

----------------------------------------

(5) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))

The TRS R consists of the following rules:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))

The TRS R consists of the following rules:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

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

(10) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B_{B_1}(x_1)) = x_1
   POL(a_{a_1}(x_1)) = 0
   POL(a_{b_1}(x_1)) = 1 + x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = 1

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

   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))


----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))

The TRS R consists of the following rules:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

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

(12) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
POL( b_{b_1}_1(x_1) ) = x_1 + 1
POL( b_{a_1}_1(x_1) ) = 2x_1
POL( a_{b_1}_1(x_1) ) = 2x_1 + 1
POL( a_{a_1}_1(x_1) ) = max{0, -2}

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

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))


----------------------------------------

(13)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

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

(14) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(15)
YES

----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))

The TRS R consists of the following rules:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

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

(17) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( B_{A_1}_1(x_1) ) = max{0, x_1 - 2}
POL( a_{b_1}_1(x_1) ) = max{0, 2x_1 - 2}
POL( b_{a_1}_1(x_1) ) = 2x_1 + 2
POL( b_{b_1}_1(x_1) ) = x_1 + 2
POL( a_{a_1}_1(x_1) ) = max{0, -2}

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

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))


----------------------------------------

(18)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))

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

(19) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(20)
YES