12.01/3.90	YES
12.01/3.91	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
12.01/3.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.01/3.91	
12.01/3.91	
12.01/3.91	Termination w.r.t. Q of the given QTRS could be proven:
12.01/3.91	
12.01/3.91	(0) QTRS
12.01/3.91	(1) QTRS Reverse [EQUIVALENT, 0 ms]
12.01/3.91	(2) QTRS
12.01/3.91	(3) DependencyPairsProof [EQUIVALENT, 16 ms]
12.01/3.91	(4) QDP
12.01/3.91	(5) QDPOrderProof [EQUIVALENT, 12 ms]
12.01/3.91	(6) QDP
12.01/3.91	(7) QDPOrderProof [EQUIVALENT, 74 ms]
12.01/3.91	(8) QDP
12.01/3.91	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
12.01/3.91	(10) TRUE
12.01/3.91	
12.01/3.91	
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(0)
12.01/3.91	Obligation:
12.01/3.91	Q restricted rewrite system:
12.01/3.91	The TRS R consists of the following rules:
12.01/3.91	
12.01/3.91	   a(a(a(x1))) -> a(a(b(x1)))
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	
12.01/3.91	Q is empty.
12.01/3.91	
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(1) QTRS Reverse (EQUIVALENT)
12.01/3.91	We applied the QTRS Reverse Processor [REVERSE].
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(2)
12.01/3.91	Obligation:
12.01/3.91	Q restricted rewrite system:
12.01/3.91	The TRS R consists of the following rules:
12.01/3.91	
12.01/3.91	   a(a(a(x1))) -> b(a(a(x1)))
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	
12.01/3.91	Q is empty.
12.01/3.91	
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(3) DependencyPairsProof (EQUIVALENT)
12.01/3.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(4)
12.01/3.91	Obligation:
12.01/3.91	Q DP problem:
12.01/3.91	The TRS P consists of the following rules:
12.01/3.91	
12.01/3.91	   A(a(a(x1))) -> B(a(a(x1)))
12.01/3.91	   B(a(b(x1))) -> A(b(a(x1)))
12.01/3.91	   B(a(b(x1))) -> B(a(x1))
12.01/3.91	   B(a(b(x1))) -> A(x1)
12.01/3.91	
12.01/3.91	The TRS R consists of the following rules:
12.01/3.91	
12.01/3.91	   a(a(a(x1))) -> b(a(a(x1)))
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	
12.01/3.91	Q is empty.
12.01/3.91	We have to consider all minimal (P,Q,R)-chains.
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(5) QDPOrderProof (EQUIVALENT)
12.01/3.91	We use the reduction pair processor [LPAR04,JAR06].
12.01/3.91	
12.01/3.91	
12.01/3.91	The following pairs can be oriented strictly and are deleted.
12.01/3.91	
12.01/3.91	   B(a(b(x1))) -> B(a(x1))
12.01/3.91	   B(a(b(x1))) -> A(x1)
12.01/3.91	The remaining pairs can at least be oriented weakly.
12.01/3.91	Used ordering:  Polynomial interpretation [POLO]:
12.01/3.91	
12.01/3.91	   POL(A(x_1)) = x_1
12.01/3.91	   POL(B(x_1)) = x_1
12.01/3.91	   POL(a(x_1)) = 1 + x_1
12.01/3.91	   POL(b(x_1)) = 1 + x_1
12.01/3.91	
12.01/3.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.01/3.91	
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	   a(a(a(x1))) -> b(a(a(x1)))
12.01/3.91	
12.01/3.91	
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(6)
12.01/3.91	Obligation:
12.01/3.91	Q DP problem:
12.01/3.91	The TRS P consists of the following rules:
12.01/3.91	
12.01/3.91	   A(a(a(x1))) -> B(a(a(x1)))
12.01/3.91	   B(a(b(x1))) -> A(b(a(x1)))
12.01/3.91	
12.01/3.91	The TRS R consists of the following rules:
12.01/3.91	
12.01/3.91	   a(a(a(x1))) -> b(a(a(x1)))
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	
12.01/3.91	Q is empty.
12.01/3.91	We have to consider all minimal (P,Q,R)-chains.
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(7) QDPOrderProof (EQUIVALENT)
12.01/3.91	We use the reduction pair processor [LPAR04,JAR06].
12.01/3.91	
12.01/3.91	
12.01/3.91	The following pairs can be oriented strictly and are deleted.
12.01/3.91	
12.01/3.91	   B(a(b(x1))) -> A(b(a(x1)))
12.01/3.91	The remaining pairs can at least be oriented weakly.
12.01/3.91	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
12.01/3.91	
12.01/3.91	   <<<
12.01/3.91	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
12.01/3.91	>>>
12.01/3.91	
12.01/3.91	   <<<
12.01/3.91	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [0A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
12.01/3.91	>>>
12.01/3.91	
12.01/3.91	   <<<
12.01/3.91	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
12.01/3.91	>>>
12.01/3.91	
12.01/3.91	   <<<
12.01/3.91	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[1A, 1A, 0A], [-I, 1A, 1A], [0A, 0A, 0A]] 	* 	x_1
12.01/3.91	>>>
12.01/3.91	
12.01/3.91	
12.01/3.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.01/3.91	
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	   a(a(a(x1))) -> b(a(a(x1)))
12.01/3.91	
12.01/3.91	
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(8)
12.01/3.91	Obligation:
12.01/3.91	Q DP problem:
12.01/3.91	The TRS P consists of the following rules:
12.01/3.91	
12.01/3.91	   A(a(a(x1))) -> B(a(a(x1)))
12.01/3.91	
12.01/3.91	The TRS R consists of the following rules:
12.01/3.91	
12.01/3.91	   a(a(a(x1))) -> b(a(a(x1)))
12.01/3.91	   b(a(b(x1))) -> a(b(a(x1)))
12.01/3.91	
12.01/3.91	Q is empty.
12.01/3.91	We have to consider all minimal (P,Q,R)-chains.
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(9) DependencyGraphProof (EQUIVALENT)
12.01/3.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
12.01/3.91	----------------------------------------
12.01/3.91	
12.01/3.91	(10)
12.01/3.91	TRUE
12.09/3.97	EOF