YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyPairsProof [EQUIVALENT, 11 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) MNOCProof [EQUIVALENT, 10 ms]
        (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 15 ms]
        (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) PisEmptyProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 27 ms]
        (18) QDP
        (19) PisEmptyProof [EQUIVALENT, 0 ms]
        (20) YES


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

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

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

Q is empty.

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

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

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(a(b(b(x1)))) -> B(b(c(c(a(a(x1))))))
   A(a(b(b(x1)))) -> B(c(c(a(a(x1)))))
   A(a(b(b(x1)))) -> A(a(x1))
   A(a(b(b(x1)))) -> A(x1)
   B(b(c(c(x1)))) -> B(b(b(b(x1))))
   B(b(c(c(x1)))) -> B(b(b(x1)))
   B(b(c(c(x1)))) -> B(b(x1))
   B(b(c(c(x1)))) -> B(x1)
   B(b(a(a(x1)))) -> A(a(c(c(b(b(x1))))))
   B(b(a(a(x1)))) -> A(c(c(b(b(x1)))))
   B(b(a(a(x1)))) -> B(b(x1))
   B(b(a(a(x1)))) -> B(x1)

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(b(c(c(x1)))) -> B(b(b(x1)))
   B(b(c(c(x1)))) -> B(b(b(b(x1))))
   B(b(c(c(x1)))) -> B(b(x1))
   B(b(c(c(x1)))) -> B(x1)
   B(b(a(a(x1)))) -> B(b(x1))
   B(b(a(a(x1)))) -> B(x1)

The TRS R consists of the following rules:

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

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

(6) UsableRulesProof (EQUIVALENT)
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.
----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(b(c(c(x1)))) -> B(b(b(x1)))
   B(b(c(c(x1)))) -> B(b(b(b(x1))))
   B(b(c(c(x1)))) -> B(b(x1))
   B(b(c(c(x1)))) -> B(x1)
   B(b(a(a(x1)))) -> B(b(x1))
   B(b(a(a(x1)))) -> B(x1)

The TRS R consists of the following rules:

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

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

(8) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

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

   B(b(c(c(x1)))) -> B(b(b(x1)))
   B(b(c(c(x1)))) -> B(b(b(b(x1))))
   B(b(c(c(x1)))) -> B(b(x1))
   B(b(c(c(x1)))) -> B(x1)
   B(b(a(a(x1)))) -> B(b(x1))
   B(b(a(a(x1)))) -> B(x1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   b(b(c(c(x0))))
   b(b(a(a(x0))))

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(a(a(x1)))) -> B(b(x1))
   B(b(a(a(x1)))) -> B(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B(x_1)) = x_1
   POL(a(x_1)) = 1 + x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = x_1

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

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


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

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

   B(b(c(c(x1)))) -> B(b(b(x1)))
   B(b(c(c(x1)))) -> B(b(b(b(x1))))
   B(b(c(c(x1)))) -> B(b(x1))
   B(b(c(c(x1)))) -> B(x1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   b(b(c(c(x0))))
   b(b(a(a(x0))))

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(c(c(x1)))) -> B(b(b(x1)))
   B(b(c(c(x1)))) -> B(b(b(b(x1))))
   B(b(c(c(x1)))) -> B(b(x1))
   B(b(c(c(x1)))) -> B(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B(x_1)) = x_1
   POL(a(x_1)) = 1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = 1 + x_1

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

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


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

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

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

The set Q consists of the following terms:

   b(b(c(c(x0))))
   b(b(a(a(x0))))

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:

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

The TRS R consists of the following rules:

   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
   b(b(a(a(x1)))) -> a(a(c(c(b(b(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.

   A(a(b(b(x1)))) -> A(x1)
   A(a(b(b(x1)))) -> A(a(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A(x_1)) = x_1
   POL(a(x_1)) = 1 + x_1
   POL(b(x_1)) = 1 + x_1
   POL(c(x_1)) = 0

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

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


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

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

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