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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 36 ms]
(8) QDP
(9) PisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( B_1(x_1) ) = max{0, x_1 - 1}
POL( b_1(x_1) ) = x_1 + 1
POL( a_1(x_1) ) = x_1 + 1
POL( c_1(x_1) ) = max{0, x_1 - 2}

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

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


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

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

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

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

(9) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(10)
YES