/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 3 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 551 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) TRUE


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

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

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

The TRS R consists of the following rules:

   b(b(x1)) -> b(a(b(x1)))
   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
   b(a(b(x1))) -> b(a(a(b(x1))))
   b(a(a(b(a(b(x1)))))) -> b(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.

   B(b(a(b(x1)))) -> B(b(x1))
   B(a(a(b(a(b(x1)))))) -> B(b(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, 0A], [0A, 1A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
>>>


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

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


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

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

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

The TRS R consists of the following rules:

   b(b(x1)) -> b(a(b(x1)))
   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
   b(a(b(x1))) -> b(a(a(b(x1))))
   b(a(a(b(a(b(x1)))))) -> b(b(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 0 SCCs with 4 less nodes.
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(6)
TRUE