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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyPairsProof [EQUIVALENT, 10 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 55 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 19 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 0 ms]
(12) QDP
(13) QDPOrderProof [EQUIVALENT, 483 ms]
(14) QDP
(15) DependencyGraphProof [EQUIVALENT, 0 ms]
(16) TRUE


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

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

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(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(l(x1)) -> L(a(x1))
   A(l(x1)) -> A(x1)
   A(c(x1)) -> C(a(x1))
   A(c(x1)) -> A(x1)
   C(a(r(x1))) -> A(x1)
   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))
   L(r(a(a(x1)))) -> A(l(c(c(c(r(x1))))))
   L(r(a(a(x1)))) -> L(c(c(c(r(x1)))))
   L(r(a(a(x1)))) -> C(c(c(r(x1))))
   L(r(a(a(x1)))) -> C(c(r(x1)))
   L(r(a(a(x1)))) -> C(r(x1))

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(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 1 SCC with 4 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))
   A(l(x1)) -> L(a(x1))
   L(r(a(a(x1)))) -> A(l(c(c(c(r(x1))))))
   A(l(x1)) -> A(x1)
   A(c(x1)) -> C(a(x1))
   C(a(r(x1))) -> A(x1)
   A(c(x1)) -> A(x1)

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))

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

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


The following pairs can be oriented strictly and are deleted.

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

   POL(A(x_1)) = 1 + x_1
   POL(C(x_1)) = x_1
   POL(L(x_1)) = 1 + x_1
   POL(a(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(l(x_1)) = x_1
   POL(r(x_1)) = 1 + x_1

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

   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))
   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))


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

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

   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))
   A(l(x1)) -> L(a(x1))
   L(r(a(a(x1)))) -> A(l(c(c(c(r(x1))))))
   A(l(x1)) -> A(x1)
   C(a(r(x1))) -> A(x1)
   A(c(x1)) -> A(x1)

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(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 1 SCC with 1 less node.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(l(x1)) -> L(a(x1))
   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))
   A(l(x1)) -> A(x1)
   A(c(x1)) -> A(x1)
   L(r(a(a(x1)))) -> A(l(c(c(c(r(x1))))))

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))

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

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


The following pairs can be oriented strictly and are deleted.

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

   POL(A(x_1)) = 1 + x_1
   POL(L(x_1)) = 1 + x_1
   POL(a(x_1)) = 1 + x_1
   POL(c(x_1)) = x_1
   POL(l(x_1)) = 1 + x_1
   POL(r(x_1)) = 1 + x_1

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

   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))
   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))


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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(l(x1)) -> L(a(x1))
   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))
   A(c(x1)) -> A(x1)

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))

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

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


The following pairs can be oriented strictly and are deleted.

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

   POL(A(x_1)) = x_1
   POL(L(x_1)) = 0
   POL(a(x_1)) = x_1
   POL(c(x_1)) = 1 + x_1
   POL(l(x_1)) = 0
   POL(r(x_1)) = 0

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

   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))
   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))


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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(l(x1)) -> L(a(x1))
   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))

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

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


The following pairs can be oriented strictly and are deleted.

   L(r(a(a(x1)))) -> A(a(l(c(c(c(r(x1)))))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO,RATPOLO]:

   POL(A(x_1)) = x_1
   POL(L(x_1)) = [1/2]x_1
   POL(a(x_1)) = [4]x_1
   POL(c(x_1)) = [1/4]x_1
   POL(l(x_1)) = [2]x_1
   POL(r(x_1)) = [1/4]
The value of delta used in the strict ordering is 3/32.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))
   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(l(x1)) -> L(a(x1))

The TRS R consists of the following rules:

   a(l(x1)) -> l(a(x1))
   a(c(x1)) -> c(a(x1))
   c(a(r(x1))) -> r(a(x1))
   l(r(a(a(x1)))) -> a(a(l(c(c(c(r(x1)))))))

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

(15) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(16)
TRUE