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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 4 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 46 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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


The following pairs can be oriented strictly and are deleted.

   A(b(x1)) -> B(a(a(a(x1))))
   A(b(x1)) -> A(a(a(x1)))
   A(b(x1)) -> A(a(x1))
   A(b(x1)) -> A(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A_1(x_1) ) = x_1 + 2
POL( B_1(x_1) ) = x_1 + 2
POL( b_1(x_1) ) = 2x_1 + 1
POL( a_1(x_1) ) = x_1
POL( c_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(x1)) -> a(a(x1))
   a(b(x1)) -> b(a(a(a(x1))))
   a(a(x1)) -> a(c(b(x1)))


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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*A(a(x1)) -> B(x1)
The graph contains the following edges 1 > 1


*B(a(x1)) -> A(a(x1))
The graph contains the following edges 1 >= 1


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(8)
YES