26.87/7.80	YES
27.38/7.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.38/7.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.38/7.88	
27.38/7.88	
27.38/7.88	Termination w.r.t. Q of the given QTRS could be proven:
27.38/7.88	
27.38/7.88	(0) QTRS
27.38/7.88	(1) QTRS Reverse [EQUIVALENT, 0 ms]
27.38/7.88	(2) QTRS
27.38/7.88	(3) DependencyPairsProof [EQUIVALENT, 13 ms]
27.38/7.88	(4) QDP
27.38/7.88	(5) QDPOrderProof [EQUIVALENT, 94 ms]
27.38/7.88	(6) QDP
27.38/7.88	(7) QDPOrderProof [EQUIVALENT, 19 ms]
27.38/7.88	(8) QDP
27.38/7.88	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
27.38/7.88	(10) TRUE
27.38/7.88	
27.38/7.88	
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(0)
27.38/7.88	Obligation:
27.38/7.88	Q restricted rewrite system:
27.38/7.88	The TRS R consists of the following rules:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> a(b(c(c(x1))))
27.38/7.88	   c(a(x1)) -> x1
27.38/7.88	   c(b(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	Q is empty.
27.38/7.88	
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(1) QTRS Reverse (EQUIVALENT)
27.38/7.88	We applied the QTRS Reverse Processor [REVERSE].
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(2)
27.38/7.88	Obligation:
27.38/7.88	Q restricted rewrite system:
27.38/7.88	The TRS R consists of the following rules:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> c(c(b(a(x1))))
27.38/7.88	   a(c(x1)) -> x1
27.38/7.88	   b(c(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	Q is empty.
27.38/7.88	
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(3) DependencyPairsProof (EQUIVALENT)
27.38/7.88	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(4)
27.38/7.88	Obligation:
27.38/7.88	Q DP problem:
27.38/7.88	The TRS P consists of the following rules:
27.38/7.88	
27.38/7.88	   A(a(x1)) -> B(a(x1))
27.38/7.88	   B(c(x1)) -> A(a(x1))
27.38/7.88	   B(c(x1)) -> A(x1)
27.38/7.88	
27.38/7.88	The TRS R consists of the following rules:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> c(c(b(a(x1))))
27.38/7.88	   a(c(x1)) -> x1
27.38/7.88	   b(c(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	Q is empty.
27.38/7.88	We have to consider all minimal (P,Q,R)-chains.
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(5) QDPOrderProof (EQUIVALENT)
27.38/7.88	We use the reduction pair processor [LPAR04,JAR06].
27.38/7.88	
27.38/7.88	
27.38/7.88	The following pairs can be oriented strictly and are deleted.
27.38/7.88	
27.38/7.88	   B(c(x1)) -> A(x1)
27.38/7.88	The remaining pairs can at least be oriented weakly.
27.38/7.88	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, -I, 0A], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [1A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	
27.38/7.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> c(c(b(a(x1))))
27.38/7.88	   a(c(x1)) -> x1
27.38/7.88	   b(c(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(6)
27.38/7.88	Obligation:
27.38/7.88	Q DP problem:
27.38/7.88	The TRS P consists of the following rules:
27.38/7.88	
27.38/7.88	   A(a(x1)) -> B(a(x1))
27.38/7.88	   B(c(x1)) -> A(a(x1))
27.38/7.88	
27.38/7.88	The TRS R consists of the following rules:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> c(c(b(a(x1))))
27.38/7.88	   a(c(x1)) -> x1
27.38/7.88	   b(c(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	Q is empty.
27.38/7.88	We have to consider all minimal (P,Q,R)-chains.
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(7) QDPOrderProof (EQUIVALENT)
27.38/7.88	We use the reduction pair processor [LPAR04,JAR06].
27.38/7.88	
27.38/7.88	
27.38/7.88	The following pairs can be oriented strictly and are deleted.
27.38/7.88	
27.38/7.88	   A(a(x1)) -> B(a(x1))
27.38/7.88	The remaining pairs can at least be oriented weakly.
27.38/7.88	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, -I, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	   <<<
27.38/7.88	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
27.38/7.88	>>>
27.38/7.88	
27.38/7.88	
27.38/7.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> c(c(b(a(x1))))
27.38/7.88	   a(c(x1)) -> x1
27.38/7.88	   b(c(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(8)
27.38/7.88	Obligation:
27.38/7.88	Q DP problem:
27.38/7.88	The TRS P consists of the following rules:
27.38/7.88	
27.38/7.88	   B(c(x1)) -> A(a(x1))
27.38/7.88	
27.38/7.88	The TRS R consists of the following rules:
27.38/7.88	
27.38/7.88	   a(a(x1)) -> c(c(b(a(x1))))
27.38/7.88	   a(c(x1)) -> x1
27.38/7.88	   b(c(x1)) -> a(a(x1))
27.38/7.88	
27.38/7.88	Q is empty.
27.38/7.88	We have to consider all minimal (P,Q,R)-chains.
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(9) DependencyGraphProof (EQUIVALENT)
27.38/7.88	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
27.38/7.88	----------------------------------------
27.38/7.88	
27.38/7.88	(10)
27.38/7.88	TRUE
27.74/11.38	EOF