31.62/9.37	YES
31.62/9.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.62/9.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.62/9.38	
31.62/9.38	
31.62/9.38	Termination w.r.t. Q of the given QTRS could be proven:
31.62/9.38	
31.62/9.38	(0) QTRS
31.62/9.38	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
31.62/9.38	(2) QDP
31.62/9.38	(3) QDPOrderProof [EQUIVALENT, 130 ms]
31.62/9.38	(4) QDP
31.62/9.38	(5) QDPOrderProof [EQUIVALENT, 0 ms]
31.62/9.38	(6) QDP
31.62/9.38	(7) QDPOrderProof [EQUIVALENT, 24 ms]
31.62/9.38	(8) QDP
31.62/9.38	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
31.62/9.38	(10) TRUE
31.62/9.38	
31.62/9.38	
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(0)
31.62/9.38	Obligation:
31.62/9.38	Q restricted rewrite system:
31.62/9.38	The TRS R consists of the following rules:
31.62/9.38	
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	
31.62/9.38	Q is empty.
31.62/9.38	
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(1) DependencyPairsProof (EQUIVALENT)
31.62/9.38	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(2)
31.62/9.38	Obligation:
31.62/9.38	Q DP problem:
31.62/9.38	The TRS P consists of the following rules:
31.62/9.38	
31.62/9.38	   A(x1) -> C(x1)
31.62/9.38	   C(a(b(x1))) -> A(a(c(x1)))
31.62/9.38	   C(a(b(x1))) -> A(c(x1))
31.62/9.38	   C(a(b(x1))) -> C(x1)
31.62/9.38	
31.62/9.38	The TRS R consists of the following rules:
31.62/9.38	
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	
31.62/9.38	Q is empty.
31.62/9.38	We have to consider all minimal (P,Q,R)-chains.
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(3) QDPOrderProof (EQUIVALENT)
31.62/9.38	We use the reduction pair processor [LPAR04,JAR06].
31.62/9.38	
31.62/9.38	
31.62/9.38	The following pairs can be oriented strictly and are deleted.
31.62/9.38	
31.62/9.38	   C(a(b(x1))) -> C(x1)
31.62/9.38	The remaining pairs can at least be oriented weakly.
31.62/9.38	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	
31.62/9.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.62/9.38	
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	
31.62/9.38	
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(4)
31.62/9.38	Obligation:
31.62/9.38	Q DP problem:
31.62/9.38	The TRS P consists of the following rules:
31.62/9.38	
31.62/9.38	   A(x1) -> C(x1)
31.62/9.38	   C(a(b(x1))) -> A(a(c(x1)))
31.62/9.38	   C(a(b(x1))) -> A(c(x1))
31.62/9.38	
31.62/9.38	The TRS R consists of the following rules:
31.62/9.38	
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	
31.62/9.38	Q is empty.
31.62/9.38	We have to consider all minimal (P,Q,R)-chains.
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(5) QDPOrderProof (EQUIVALENT)
31.62/9.38	We use the reduction pair processor [LPAR04,JAR06].
31.62/9.38	
31.62/9.38	
31.62/9.38	The following pairs can be oriented strictly and are deleted.
31.62/9.38	
31.62/9.38	   C(a(b(x1))) -> A(c(x1))
31.62/9.38	The remaining pairs can at least be oriented weakly.
31.62/9.38	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	
31.62/9.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.62/9.38	
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	
31.62/9.38	
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(6)
31.62/9.38	Obligation:
31.62/9.38	Q DP problem:
31.62/9.38	The TRS P consists of the following rules:
31.62/9.38	
31.62/9.38	   A(x1) -> C(x1)
31.62/9.38	   C(a(b(x1))) -> A(a(c(x1)))
31.62/9.38	
31.62/9.38	The TRS R consists of the following rules:
31.62/9.38	
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	
31.62/9.38	Q is empty.
31.62/9.38	We have to consider all minimal (P,Q,R)-chains.
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(7) QDPOrderProof (EQUIVALENT)
31.62/9.38	We use the reduction pair processor [LPAR04,JAR06].
31.62/9.38	
31.62/9.38	
31.62/9.38	The following pairs can be oriented strictly and are deleted.
31.62/9.38	
31.62/9.38	   C(a(b(x1))) -> A(a(c(x1)))
31.62/9.38	The remaining pairs can at least be oriented weakly.
31.62/9.38	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[1A, 1A, 1A], [1A, 1A, -I], [-I, -I, 1A]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, -I], [-I, -I, -I]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	   <<<
31.62/9.38	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
31.62/9.38	>>>
31.62/9.38	
31.62/9.38	
31.62/9.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.62/9.38	
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	
31.62/9.38	
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(8)
31.62/9.38	Obligation:
31.62/9.38	Q DP problem:
31.62/9.38	The TRS P consists of the following rules:
31.62/9.38	
31.62/9.38	   A(x1) -> C(x1)
31.62/9.38	
31.62/9.38	The TRS R consists of the following rules:
31.62/9.38	
31.62/9.38	   a(x1) -> b(b(c(x1)))
31.62/9.38	   c(a(b(x1))) -> a(a(c(x1)))
31.62/9.38	   c(b(x1)) -> x1
31.62/9.38	
31.62/9.38	Q is empty.
31.62/9.38	We have to consider all minimal (P,Q,R)-chains.
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(9) DependencyGraphProof (EQUIVALENT)
31.62/9.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
31.62/9.38	----------------------------------------
31.62/9.38	
31.62/9.38	(10)
31.62/9.38	TRUE
32.75/9.44	EOF