/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 22 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 1 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 7 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) TRUE


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

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

Q is empty.

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(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:

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

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = 15 + x_1
   POL(b(x_1)) = 17 + x_1
   POL(c(x_1)) = 9 + x_1
   POL(d(x_1)) = 21 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(x1)) -> c(d(x1))
   d(c(x1)) -> a(a(x1))

Q is empty.

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(5) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(a(x1)) -> D(x1)
   D(c(x1)) -> A(a(x1))
   D(c(x1)) -> A(x1)

The TRS R consists of the following rules:

   a(a(x1)) -> c(d(x1))
   d(c(x1)) -> a(a(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   D(c(x1)) -> A(a(x1))
   D(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(D(x_1)) = 1 + x_1
   POL(a(x_1)) = 1 + x_1
   POL(c(x_1)) = 1 + x_1
   POL(d(x_1)) = 1 + x_1

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

   a(a(x1)) -> c(d(x1))
   d(c(x1)) -> a(a(x1))


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

   A(a(x1)) -> D(x1)

The TRS R consists of the following rules:

   a(a(x1)) -> c(d(x1))
   d(c(x1)) -> a(a(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
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(10)
TRUE