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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 77 ms]
(4) QDP
(5) PisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

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

Q is empty.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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


The following pairs can be oriented strictly and are deleted.

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

   POL(A(x_1)) = [7/4] + [2]x_1
   POL(B(x_1)) = [1/2]x_1
   POL(a(x_1)) = [3/2] + [2]x_1
   POL(b(x_1)) = [1/2]x_1
   POL(c(x_1)) = [4] + [2]x_1
The value of delta used in the strict ordering is 1/4.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(6)
YES