31.39/8.87	YES
31.39/8.90	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.39/8.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.39/8.90	
31.39/8.90	
31.39/8.90	Termination w.r.t. Q of the given QTRS could be proven:
31.39/8.90	
31.39/8.90	(0) QTRS
31.39/8.90	(1) QTRS Reverse [EQUIVALENT, 0 ms]
31.39/8.90	(2) QTRS
31.39/8.90	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
31.39/8.90	(4) QDP
31.39/8.90	(5) QDPOrderProof [EQUIVALENT, 131 ms]
31.39/8.90	(6) QDP
31.39/8.90	(7) QDPOrderProof [EQUIVALENT, 1000 ms]
31.39/8.90	(8) QDP
31.39/8.90	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
31.39/8.90	(10) TRUE
31.39/8.90	
31.39/8.90	
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(0)
31.39/8.90	Obligation:
31.39/8.90	Q restricted rewrite system:
31.39/8.90	The TRS R consists of the following rules:
31.39/8.90	
31.39/8.90	   b(b(b(b(x1)))) -> b(b(b(a(b(x1)))))
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	Q is empty.
31.39/8.90	
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(1) QTRS Reverse (EQUIVALENT)
31.39/8.90	We applied the QTRS Reverse Processor [REVERSE].
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(2)
31.39/8.90	Obligation:
31.39/8.90	Q restricted rewrite system:
31.39/8.90	The TRS R consists of the following rules:
31.39/8.90	
31.39/8.90	   b(b(b(b(x1)))) -> b(a(b(b(b(x1)))))
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	Q is empty.
31.39/8.90	
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(3) DependencyPairsProof (EQUIVALENT)
31.39/8.90	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(4)
31.39/8.90	Obligation:
31.39/8.90	Q DP problem:
31.39/8.90	The TRS P consists of the following rules:
31.39/8.90	
31.39/8.90	   B(b(b(b(x1)))) -> B(a(b(b(b(x1)))))
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(b(a(b(b(x1)))))
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(a(b(b(x1))))
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(b(x1))
31.39/8.90	
31.39/8.90	The TRS R consists of the following rules:
31.39/8.90	
31.39/8.90	   b(b(b(b(x1)))) -> b(a(b(b(b(x1)))))
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	Q is empty.
31.39/8.90	We have to consider all minimal (P,Q,R)-chains.
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(5) QDPOrderProof (EQUIVALENT)
31.39/8.90	We use the reduction pair processor [LPAR04,JAR06].
31.39/8.90	
31.39/8.90	
31.39/8.90	The following pairs can be oriented strictly and are deleted.
31.39/8.90	
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(a(b(b(x1))))
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(b(x1))
31.39/8.90	The remaining pairs can at least be oriented weakly.
31.39/8.90	Used ordering:  Polynomial interpretation [POLO]:
31.39/8.90	
31.39/8.90	   POL(B(x_1)) = 2*x_1
31.39/8.90	   POL(a(x_1)) = x_1
31.39/8.90	   POL(b(x_1)) = 1 + x_1
31.39/8.90	
31.39/8.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.39/8.90	
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	   b(b(b(b(x1)))) -> b(a(b(b(b(x1)))))
31.39/8.90	
31.39/8.90	
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(6)
31.39/8.90	Obligation:
31.39/8.90	Q DP problem:
31.39/8.90	The TRS P consists of the following rules:
31.39/8.90	
31.39/8.90	   B(b(b(b(x1)))) -> B(a(b(b(b(x1)))))
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	The TRS R consists of the following rules:
31.39/8.90	
31.39/8.90	   b(b(b(b(x1)))) -> b(a(b(b(b(x1)))))
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	Q is empty.
31.39/8.90	We have to consider all minimal (P,Q,R)-chains.
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(7) QDPOrderProof (EQUIVALENT)
31.39/8.90	We use the reduction pair processor [LPAR04,JAR06].
31.39/8.90	
31.39/8.90	
31.39/8.90	The following pairs can be oriented strictly and are deleted.
31.39/8.90	
31.39/8.90	   B(b(b(b(x1)))) -> B(a(b(b(b(x1)))))
31.39/8.90	The remaining pairs can at least be oriented weakly.
31.39/8.90	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.39/8.90	
31.39/8.90	   <<<
31.39/8.90	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
31.39/8.90	>>>
31.39/8.90	
31.39/8.90	   <<<
31.39/8.90	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
31.39/8.90	>>>
31.39/8.90	
31.39/8.90	   <<<
31.39/8.90	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 1A, 0A], [-I, 0A, 1A], [-I, 1A, 0A]] 	* 	x_1
31.39/8.90	>>>
31.39/8.90	
31.39/8.90	
31.39/8.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.39/8.90	
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	   b(b(b(b(x1)))) -> b(a(b(b(b(x1)))))
31.39/8.90	
31.39/8.90	
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(8)
31.39/8.90	Obligation:
31.39/8.90	Q DP problem:
31.39/8.90	The TRS P consists of the following rules:
31.39/8.90	
31.39/8.90	   B(a(b(b(a(b(x1)))))) -> B(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	The TRS R consists of the following rules:
31.39/8.90	
31.39/8.90	   b(b(b(b(x1)))) -> b(a(b(b(b(x1)))))
31.39/8.90	   b(a(b(b(a(b(x1)))))) -> b(b(a(b(b(x1)))))
31.39/8.90	
31.39/8.90	Q is empty.
31.39/8.90	We have to consider all minimal (P,Q,R)-chains.
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(9) DependencyGraphProof (EQUIVALENT)
31.39/8.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
31.39/8.90	----------------------------------------
31.39/8.90	
31.39/8.90	(10)
31.39/8.90	TRUE
31.66/8.96	EOF